Interesting question. As Matthew Clifford said, normal limits for skyscrapers don’t apply. But there are some limitations even so. I’m going to take a slightly different slant on it. It can certainly grow to be kilometers in diameter no problem. But exactly how large can it get?

First, in case any of you don’t know, the current plan is to de-orbit it in the mid 2020s. Most of it will crash into the Pacific in a region known as the spacecraft graveyard, an area far from any shipping route and with deep sea where many burnt up spacecraft lie on the ocean floor. Even if they had the funding to keep it going, the modules themselves have a short lifetime, because the conditions are so harsh there. There are a couple of new modules which could be kept and used as part of a new space station - and that’s a possible Russian plan, the new ones are theirs. But the rest will go.

So, this is hypothetical - as an imaginative exercise.

BUILDING VERTICALLY - OR SIDEWAYS - CAN GET TO KILOMETERS, MAYBE EVEN HUNDREDS OF KILOMETERS

So, first, if you build vertically, you are going to run into problems eventually. Satellites closer to Earth orbit faster. The gravity also changes so there’s a gravity gradient as well. LEO is well within Earth’s “Roche sphere”. This means that a structure even made of solid rock will tear apart into dust if it is big enough. Tides from the Moon are relevant too.

So, you will certainly be able to build hundreds of meters, and probably kilometers. You can get to tens, maybe hundreds of kilometers if you join the modules together with materials such as Kevlar, or better, Zylon (polybenzoxazole fiber) which has a breaking height of 379 km under full Earth gravity (from page 14 of this study). When a material has a breaking height, you can build cables that can support their own weight for several times that breaking height if you taper them.

If we can build our structures with carbon monotubes or similar - which we don’t have yet of that length, eventually we could build it tens of thousands of kilometers vertically all the way to GEO and beyond. The problem with carbon nanotubes is that minute flaws destroy just about all their strength, Perfect carbon nanotubes measured in the laboratory have a breaking height of 2,200 km, but all the carbon nanotubes constructed for practical applications have only a hundredth of their theoretical strength. They are actually weaker than Kevlar (1 GPa compared to 3.6 GPa for Kevlar).

Maybe one day we can do that. But right now you are probably limited to a few hundred kilometers of vertical building at most, especially in LEO where the gravity gradient is high. At higher orbits, then you could build it vertically much further than at lower orbits because the gravity gradient is much less

You could use air beams to connect the modules together as they would permit more flexibility in such a large structure. That’s plenty of expansion space for the foreseeable near future. But I like to explore the limits of what is possible. So suppose you want it to be even bigger, not just hundreds, but thousands of kilometers in dimension?

It’s also an issue building sideways too. Again you run into the problem that different parts of your space station will try to move in different directions around the Earth because the orbits aren’t parallel, indeed they intersect.

BUILDING ALONG ITS ORBIT - NO PROBLEM AT ALL - TENS OF THOUSANDS OF KILOMETERS

So, by far the easiest way to build is along its orbit. It could be joined with other modules to fill its entire orbit. That would be no problem as they are all orbiting with the same orbital velocity.

So the best way to build it really big is to make it - perhaps a few hundred meters wide and deep, or kilometers if you need to, and as a huge ribbon that spreads around the world, so that would make it 43,000 kilometers long.

That may seem impossible. But we may get low cost transport to space in the future, and if so, then there isn’t really any reason why we can’t continue to add to it.

ORBITAL DRAG AT LEO

Now there is the problem that it is a bit too low however to join together like that, because it will de-orbit within decades at most, probably faster. You wouldn’t want to build a huge orbital metropolis and then reboost it every few years. So it would make more sense to build it at a rather higher orbit, middle orbit. Or you could go rather higher and build it in geostationary orbit.

ORBITAL CITY AT GEOSTATIONARY ORBIT

Geostationary orbit is 35,786 kilometres above the equator, add the radius of the Earth of 6,371 km and you get a circumference of around 264,000 km. So if we were to built at GEO we could have something like the ISS that’s truly vast. It’s the next most natural place to build after LEO in a future with abundant heavy lift and faster rockets, making it easy perhaps to get even to GEO in hours.

If you build in GEO also, then there is nothing there to make it de-orbit. It will just stay there indefinitely.

Arthur C. Clarke envisions such a future in his Fountains of Paradise novel, in the epilogue “Kalidasa’s triumph”

“The Holmer everted Its eyes to give telescopic vision, and slowly scanned the zenith. Yes, there it was - hard to see by day, but easy by night when the sunlight streaming past the shadow of Earth still blazed upon it. The thin, shining band that split the sky into two hemispheres was a whole world in itself, where half - a - billion humans had opted for permanent zero gravity life.”

The Fountains Of Paradise

So how big could such a city be? What surface area? Well the surface area of the Earth is 510.1 million km². Suppose our city is 100 km wide. Then how many levels would be needed to achieve the same surface area as the Earth?

The answer is that if you made it 20 stories high, 100 km wide, and filled the whole of GEO with it, then the surface area inside would be larger than the surface area of our Earth. You could make each of those “stories” kilometers high and it would still be feasible. Indeed the gravity gradient there is so little you could build it vertically for thousands of kilometers easily if you had any need to do so.

If we had the ability to move large amounts of materials into space including getting construction materials from the Moon, and if we suppose a civilization continuing, space faring, for a few millennia, then I can easily imagine that we produce constructions like this eventually.

See also my answer to Why is it so difficult to penetrate our atmosphere with a returning spacecraft? In other words, why can’t the vehicle slowly enter our atmosphere?