I think we need to take tremendous care here. Before I go into the rest of this answer - it may seem "pessimistic" but I think that this is a future that won't happen, because any sensible ET will make sure it can't happen. The question is how we will prevent it, that's all. I read it the other way, that because such a future can't be permitted to happen, and that if we look at this carefully everyone will agree that it must not happen - that we will find a way to explore the galaxy without trashing it and without these dire consequences for ourselves and everyone else in the galaxy. But how, I don't know yet. What we can do though is to start off with safe exploration which is to say, send robots but not attempt human colonization of other stars, not until we see a way through this.
UNCONTROLLED REPLICATION AND THE PAPER CLIP EVENT HORIZON
There's been a lot of talk about the effect uncontrolled self replicating machines could have on the galaxy if they got out of hand. Someone makes a paper clip making machine that self replicates and is programmed in some clever way to evolve to get better and better at making paperclips. But very carefully designed so it always makes 99% paperclips, say. Their idea is just to seed a region of space with these machines, go away, come back and scoop up all the paperclips and sell them.
But - the machines get too good at it, so good that when they come to scoop them up, the machines instantly turn all their spaceships into paperclips as well. And then head off and turn the entire galaxy into paperclips. Not for any particular reason. They have no motive, it's just following their programming. And because of their clever programming they get better at what they do until they are completely unstoppable.
That's the "paper clip event horizon".
WHAT ARE HUMANS - IF NOT UNCONTROLLED SELF REPLICATORS?
But - what are humans except uncontrolled self replicators? We can't be as strictly programmed as machines. But is that a benefit? Might it not be a hazard instead?
What is to stop the humans from making self replicators once they get out of sight of everyone else? Or becoming terrors of the galaxy themselves?
Something that might lead to particular caution here is if you look at how the mechanism of evolution might play out on a galaxy sized canvas.
Surely most colonists spreading through the galaxy would behave responsibly. But- look what happens if some group of colonists starts to really encourage rapid population growth? Or cloning?
As with the mechanical self replicators, the ones that manage to make most copies of themselves, so long as it is consistent with continuing to colonize the galaxy, will be the winners of this race to colonize the entire galaxy.
The youngest human to give birth is a young girl of 5. So if there is a drive towards more and more rapid population growth, then children might give birth at age 5 without needing to do much evolution at all beyond what we are like right now. Or hundreds of clones of each person. This would become the horrific norm in the most rapidly expanding population.
Any population that did that would be the one that spreads out and occupies the galaxy. So long as it was also able to maintain space technology while doing so.
CAN'T EXPAND EXPONENTIALLY FOR LONG AT ALL
With a doubling every ten years, you could get to as many humans in a century as you could normally get to in a thousand years. Starting with a population like that of the present day Earth - then within 200 years you need to find homes for the entire current population of the Earth twice a week.
Once you have an entire solar system filled with both types of humans - then if they are able to keep going - for every new solar system filled with humans that double every 100 years, there would be over a thousand new solar systems filled with humans that double every ten years.
As time goes on that disparity would just get larger and larger until the ones that double only every 100 years become all but extinct. Especially if the faster doubling humans are more aggressive and warlike as well and actively destroy the slower doubling rate humans and take over their colonies for their expanding populations.
If you can manage to fill an entire solar system with trillions of colonists within a couple of centuries, and fill thousands of solar systems within a century, then you'll win this race to colonize the galaxy.
And then, the ones that are like ISIS destroying wherever they go, again would seem likely to be favoured in this race. Because soon they would have no competition - they would be the only ones left.
And - the thing is that the exponential growth also has to stop at some point.
If we double in population every century, then by ten thousand years from now, doubling every century, you need to convert at least the sun's mass into people every century to keep going - that's supposing dwarf humans of only 5 kg each, same size as the smallest man in the world. And by 13,000 years, you convert an entire galaxy into humans every century - assuming that you have Faster Than Light travel.
But given that the process would favour very rapid doubling, maybe down to doubling every five years or less, those numbers would reduce to 650 years from the first moment that two human colonists achieve doubling every five years to the point where their progeny need to convert an entire galactic mass into humans every five years.
If they haven't achieved FTL of course they hit the crisis much sooner, also it all happens so quickly probably they haven't got the technology yet to convert stars into humans so quickly.
Compared with the age of the galaxy, this time limit of at most 13, 000 years (for one century doubling time and FTL and convert an entire galaxy mass into humans) and quite possibly as short as less than 1,000 years before crisis (faster doubling time) is just a blink of time.
IN NATURE EXPONENTIAL GROWTH OFTEN STOPS WITH A CRASH - LOW LIFE EXPECTANCY OR POPULATION PLUMMETS
And the thing is - that yes of course in nature this sort of thing often happens. But it gets settled by the population dying off in huge numbers. Can be a steady state where most of the progeny die very young.
Or could be a boom and crash like lemmings.
Either way - it's not something as intelligent creatures you'd want to have in your future for your own population. Remember, with a doubling time of five years, that means that half your population has to die every five years, one way or another, either continuously or by boom and bust, throughout the galaxy, in your steady state situation.
And with no possibility of communicating galaxy wide to find a solution, and with the most aggressive and the ones with the best technology favoured in all the clashes, it seems likely that once it got to that point, that this situation of continual either warfare or starvation would just continue on and on for the future.
WON'T MAKE THEMSELVES EXTINCT
So long as the ETs are confined to a single solar system you can imagine them going extinct through recklessness. But once they start to colonize a galaxy - even if many of the colonies go extinct, hard to see how that could happen to them all. Just needs one of the millions of star systems to have a single couple that still has the original vigour of the initial population expansion - and after an extinction event they'd quickly repopulate all the surrounding worlds. It would just be this scenario on and on into the foreseeable future, seems to me. Unless some other species took over from them. But the idea that it might just stop, and be left with an uninhabited galaxy with relics of an ancient galaxy wide civilization - again I can't really see that in reality though it is fun in science fiction - at least not without FTL.
CURRENTLY WE SEEM TO BE HEADING TOWARDS A MUCH MORE PEACEFUL GENTLE SLOW DOWN THAN THAT
We seem to be heading towards stability of our own population, at least is a reasonable chance, by peaceful methods, without anything like this happening.
Middle of road and low projections have our population stabilizing or even decreasing.
So, hopefully we will avoid the horrors of unrestricted exponential growth. It seems promising at least that our birth rate will just slow down naturally.
When affluent, then parents choose to have fewer children and devote more effort on bringing them up, rather than to have large families. Coupled with the ones who don't have children that leads to steady state or even a drop in population in all the wealthiest countries of all religions and philosophies. And as a result we have now reached peak child, where no more children were born this year than several years ago. Continue to have an exponential growth curve because people are living longer and average life span is still increasing rapidly in the poorer countries especially.
So that is reasonably promising for Earth. Lots of major challenges, but it could have been a whole lot worse than this.
THIS THOUGH SEEMS TO DEPEND ON AN EXPANSION INTO A LIMITED SPACE
So, that works fine if limited to the Earth. Or hopefully will. Even if limited to our solar system I can imagine something like that working, with ability to work through problems together.
But it's hard to see this sort of natural peaceful solution happening any more once you have galactic colonization, because that so strongly favours the few who have rapidly increasing populations.
ONLY THE TINIEST PERCENTAGE NEEDS TO BE EXPANSIONIST TO START WITH
Remember - I'm not at all suggesting that most or even many of the colonists woul do this. But even if 99.99999% of colonists are peaceful and non expansionist, the remaining tiny fraction of a percent would be the ones that keep setting off to settle new solar systems. Once you have thousands of trillions of colonists then probably you get new colonization ventures every week or day like that. And some of those would be bound to be more expansionist and favouring faster growing populations than others - and those are the ones that would win the race.
The likes of ISIS, the Nazis, the worst types of society our world has seen - and far worse - if it can happen on Earth it can surely happen in an expanding wave of trillions of colonists where anyone with a bit of capital can set off and start a new colony around a new star system. The best also of course. But the point is that in this wave of colonization it would seem, with trillions of colonists, that the most aggressive and expansionist would be favoured. Because, unlike the situation on Earth, there are no geographic restrictions to hold them back and force them to find a compromise with their neighbours.
What would the Nazis have done if instead of expanding into a world already colonized by many other humans they were expanding into a galaxy of uninhabited solar systems? They'd be impossible to defeat because they could keep retreating into more distant parts of the galaxy, then return again whenever they felt the time was right to apply their "final solution" to the rest of us. And by then they would have technology that they have developed in isolation not just for a few years, but secretly for centuries or thousands of years. Here in our world, they already developed missile technology way ahead of anyone else, the basis for the moon rockets soon after the war finished. It would be like the fictional future world in The Man in the High Castle - but galaxy spanning and with nightmare technology developed over centuries.
They don't need to have horrific ideologies. They could just be very good at technology and rapidly expanding populations. The ones with the fastest growing populations ,whatever their ideologies, would win the race.
The winners needn't even be human. Could be cyborg, part human. Or be animals with their DNA adapted - for uplift - which probably can be a good thing in the right hands as in David Brin's novels - but the worst ones would be the ones that spread rapidly.
Once more not saying at all that the majority of colonists or indeed hardly any at all would be like this initially. But you'd have to be mighty sure that nothing like this could happen later on.
NO NEED TO STOP TO TERRAFORM
And - if they had to terraform planets to keep expanding, that would slow things down also. But - no need to suppose they would. Terraforming seems likely to be very hard to do, maybe impossible. And far easier to just make new habitats from the asteroid belts and comets. Not that far off from doing it now. With a century or two of development of technology it might be as easy to set up home in the asteroid belt or the Oort cloud (using fusion power for eneregy) as on Earth. Once that happens, you can migrate constantly anywhere throughout the galaxy.
Once you have a galaxy or a large region filled by rapidly doubling populations that are also by habit aggressive, they will surely turn on each other - how can it be stopped? With no way to engage in diplomacy, because of light speed barriers to communication, no way to even meet up with them or know they exist until they attack you, how can we do anything about it?
BEYOND ANY POSSIBILITY OF COMMUNICATING WITH THEM AND TRYING TO STOP THEM
And - the rapidly growing ones would probably also have the better technology too. And what makes it really dangerous is that it would happen beyond the light speed horizon. The most expansionist colonists would end up a thousand light years away within a few millennia.
If you send an expedition or message to try to stop them, it takes a thousand years to get to them by which time they have moved forward yet another thousand years in their technology.
As it is now, if we have a dangerous element of our society, at least they are contained. They are on the same world as us. One way or another eventually it has to get sorted out. But in this ever expanding colonization - once interstellar colonization gets easy - they just disappear beyond the light speed horizon. With a galaxy 100,000 years in diameter, then - soon you get colonists ten years away, then a century, then a thousand light years away.
They then take over millions of solar systems, develop their technology way beyond the rest of us, and then come back out of the night sky rampaging over all the peaceful slow colonizing parties.
They'd be able to do that because with such huge populations, they have much more opportunity for innovation and would have better technology. And with such rapid population growth, they'd be desperate people too. Already completely filled many solar systems to the point that they have run out of resources there, for their vast populations, the few solar systems not filled by their kind would seem easy pickings.
TECHNOLOGY SO AHEAD OF US IT IS INVISIBLE TO US
They might have technology that is completely invisible to us, arrives as nanoparticles traveling close to speed of light resembling cosmic rays, hit into the Oort cloud objects to slow down, spread surreptitiously, and then take over a solar system and converts it, say, into a Dyson sphere or whatever, paying no attention to whether any of the planets is already inhabited.
For this reason I think we simply shouldn't dream of sending human colonists out into the galaxy until we have worked this through.
There may be ways of colonizing peacefully and safely. And surely mature ET civilizations wouldn't behave like this - it would be so dangerous for themselves also apart from anything else. But I can see humans doing it in the near future, if we colonize a galaxy without first maturing somewhat first.
NEED TO WORK IT OUT FIRST, SO WE DON'T BECOME MONSTERS OF THE GALAXY - AND FOR OURSELVES ALSO
So, we need to work it out before we colonize, make sure we aren't going to end up in this nightmare scenario where we are monsters for ourselves - and also for other ETs too. Because the galaxy may be filled with many non technological ETs for every technological one. Of the creatures close to human intelligence on our world:
Octopus - dextrous but hard to see how it could develop fire and metal working
Elephant - quite dextrous but could it really do everythngn we do with just its trunk and feet? Could an elephant light a fire, or make a stone axe?
Grey Parrots - perhaps the most intelligent of all. Quite dextrous. But they are so tiny, how far could they get being so feeble and small? Maybe they would grow big like ostriches
Dolphins - hard to see how they could evolve any significant technology at all.
Primates - of course they have a good chance once they develop opposable thumbs.
And then consider how there may be many water worlds especially if you included oceans of icy moons.
ADVANCED TECHNOLOGICAL ETIS WOULD PROBABLY SEE GALACTIC COLONIZATION OF THIS TYPE AS A CANCER TO BE PREVENTED
So - well if our colonists encounter other technological civilizations - they've got a chance, millions of years ahead of us probably. But in that case, they might see us as a cancer and have robot scouts to detect things like this and nip it in the bud early on. Perhaps by sterilizing us in some irreversible way until our population is low enough again?
But what if we are first, as in your scenario? Either first intelligent or else - first technological ET, and the others are non technological. That would be worst of all. Nobody to learn from.
SURELY ETIS IF SENSIBLE WILL STOP AT THIS POINT WHERE EXPONENTIAL GROWTH INTO A GALAXY IS TECHNOLOGICALY POSSIBLE
It's become my preferred explanation for the "great silence" and Fermi's paradox. I think any sensible ET, once they get to the point where they could, potentially, colonize the galaxy - will stop at this point and not rush out, but think things through before continuing.
Unlike animals like Lemmings, we can look at future consequences and make decisions, not based on blind evolution. We already have done many such. E.g. there probably wouldn't be a single blue whale left if humans only operated on blind evolution without ability for forethought and understanding and planning.
And the ones who are not sensible enough to do that are probably not sensible enough to set up large scale colonies in a solar system either - because space colonies would be so fragile, and easily destroyed, surrounded by vacuum and with such powerful technologies available to the colonists. They have to be more peaceful than we are at present, and forward looking, to have significant numbers of colonists in space to start with. Don't have to make themselves extinct, but probably keep destroying their space colonies until the manage to take the next step, whatever it is, that lets them be forward thinking and able to settle things relatively amicably.
Either that - or if they keep to relatively small numbers in space, like an Antarctic colony etc, again then can see space exploration continuing peacefully, even as we are now. But if you have entire countries of millions of people in space, it is a bit hard to see that working until we find some more peaceful way to go forward, because they would be so fragile and the technology they have so powerful.
WHATEVER THE NEXT STEP - THAT'S PROBABLY WHAT ETS DO ALSO IF THEY EXIST - PROBABLY MANY POSSIBLE SOLUTIONS
And then whatever that next step is - that would then be what ETs do, maybe lots of solutions - but whatever they are - they surely don't involve totally filling the galaxy with exponentially growing self destructive populations of ETs.
And - another way of looking at it. Populations simply can't keep growing exponentially for more than a few millenium - unless you have a very very long life, say you become fertile at age a hundred million or a billion. If you have reasonably early age of fertility - you simply can't have an exponential population for more than a few thousand years.
SIZE OF LARGEST ET POPULATION
So any ET we encounter is almost bound to be in a steady state and not currently growing in population. It doesn't make much difference if they do exponential growth in fits and starts. Still, the chance we spot an ET in the process of exponential growth right now must be tiny.
And we will also be in a steady state in the future also. Even if we did galactic colonization, at some point our population would have to stop growing because continuing exponential growth is impossible for more than a few thousand years.
So what is the size of the largest ET population in our galaxy? It's obviously not filling the entire galaxy. It might be a single solar system, just one world, a small group of a few solar systems.
But I think because of this issue of population growth, it's going to be a civilization where they are all able to keep in easy contact with each other. Which unless they have FTL probably means just spanning a few light years at most. Could have non reproducing explorers going out on long expeditions, but the core population, the ones that reproduce, I'd expect to be compact spatially in our galaxy.
If so, could be thousands of such civilizations and we wouldn't know about them yet. Their explorers could be crisscrossing our galaxy and we might be due a visit some time in the next few thousand years.
About the only way I can think of that such a civilization could have galaxy spanning colonies (rather than explorers) - apart from FTL again, is if they have immensely long lives. Long enough so that you can travel across the galaxy to see your cousin a hundred thousand light years away, travel back again (probably a million year journey either way) and think nothing of it. And stable, slow pace of change, no worry that any of your cousins would start reproducing rapidly or anything like that.
The galaxy could however be filled by their robots. Because - almost the opposite of the way it is usually presented - I think self replicating robots, carefully programmed, are actually safer than self replicating humans. Because you can do all sorts of things to make them safe that would be either impossible or at least totally unethical for humans.
Of course there could be other solutions. But I think one way or another, any ET has to think this through and not just go out and "take the galaxy" but think over what the consequences would be of doing this.
A maximum future population of course doesn't mean at all a limit to exploration of new frontiers or limits to understanding or quality of life or insights.
Actually a smaller population makes the frontiers much more endless if anything. You could explore the galaxy for millions of years and continue to find places and phenomena - and maybe new creatures and beings and other intelligent species (especially non technological) that no human had ever encountered before.
It also goes into ideas of robotic exploration and ways that robots could be made safe self replicating explorers of a galaxy, using techniques that would be totally unethical for human colonists, if possible at all.
I think our best first step forward would be careful robotic exploration. Eventually try self replicating star ship robots - but to start with, limited, maybe to our solar system, then to say within ten or a hundred light years of Earth - etc - while we work on the technology. And then meanwhile keep our Earth population contained either on Earth or in our solar system, while we find out more and figure out what to do next.
In this future, letting off an uncontrolled wave of human colonies into the galaxy may be seen as worse than an uncontrolled wave of unrestricted robotic replicators, and both would be far worse than letting off a nuclear bomb in a city. For similar reasons that it's an uncontrolled chain reaction with potentially devastating reactions - you wouldn't start up the chain reaction for a nuclear bomb hoping optimistically that this time it wouldn't explode. You just don't do it at all.
Would be so unthinkable, nobody would do it, from a young child you'd learn that this simply can't be accepted. And the few crazy people who might try to do it anyway would be restrained by everyone else and the technology developed to make it next to impossible.