Well I'd have to say, exploring artificial gravity. We have been technically capable, since the 1960s, of sending a couple of spacecraft to orbit, join them with a tether, and set them spinning around the common centre of gravity to create artificial gravity. Yet nobody has ever attempted it.
Was one try with Gemini, where they did a very slow gyration, too slow for any noticeable gravity. Another time another Gemini went out of control and tumbled for a few minutes. The skylab astronauts ran around their jogging track (but nobody seems to have done any measurements to see effect on their health). Experiments with the rotating "litter chair" in Skylab - but these were to do with understanding space sickness. Though an outcome of them was that humans seem much more tolerant of spinning motions in space than anyone expected, nobody tried to apply that to a short arm centrifuge and artificial gravity to see what happened in that situation. Experiments with rats done by the Russians that showed that artificial gravity in a centrifuge helped them to keep healthy in space - but never tried on a human (rats can't get sick so you can't directly use the results to predict how humans would respond).
It would be so easy to do. Joe Carroll came up with a way we could do it with a routine crewed Soyuz mission to the ISS without using any extra fuel even, very ingenious approach, using the spent third stage as the counterweight which is usually discarded.
But nobody seems to be the slightest bit interested amongst those who make decisions about these things. Zero interest in it now for well over 40 years. Although before the first space flights there were many ideas for spinning space stations with artificial gravity, and many ideas since then for tethers for gravity, and for spinning centrifuge sleeping quarters, and all sorts, but they just remain paper studies.
You might think - that there is nothing much to learn as we know how fast to spin and what length of tether to get any desired amount of artificial gravity.
But the thing is - that we know nothing about how humans would respond to those spins. It is very different situation from spinning in a centrifuge on the ground.
On the ground you always have gravity acting along the spin axis - no gravity acting along the spin axis in space. This makes a difference to parts of our inner ear that make us feel giddy.
On the ground it is a turning motion. In space it is a tumbling motion - stimulates a different part of our inner ear.
.On the ground, you feel the coriolis effect when you move your hand sideways or try to walk in a straight line. In space, because it is a tumbling motion, you feel it when you move your hand up and down or stand up or sit down.
And we just can't duplicate the space conditions on the ground for more than a few minutes in an airplane.
So - we have almost no data on what happens to the human body in a spinning habitat in space. Or about whether we would feel dizzy or nauseous or not notice it at all. Or whether we would adapt to it over severl days or not.
We also know nothing at all about the effect of it on the human body. Would an hour a day of full gravity in a small arm centrifuge be enough for health? Or would it work better to try, say, lunar gravity 24/7 in a slowly spinning pair of tethered spacecraft? Or would that make you actually sicker (either of them)?
So many experiments we could do, and just to do one or two of them would give us some ground truth. And we could have done this any time since the late 1960s. And it might, possibly, have totally revolutionized space travel. For instance what if a small 4 meter diameter centrifuge spinning slowly for a few hours a day is enough to keep an astronaut healthy? So healthy they don't even notice anything on return to Earth?
Just suppose that works, and achieves perfect health just as if you were still on Earth. Think how silly everyone will feel, that for several decades the astronauts have been doing these exercises, two hours or more a day, trying to reverse the effects of zero g, and still not succeeding totally when this simple idea might have made all that unnecessary?
I don't know how likely that is, but seems, at least, not something you can rule out until you try it out. Why has nobody ever tried this simple experiment?
Why never tried a short arm centrifuge in space? Why never tried tehering a spaceship to its final stage or another spaceship and setting them spinning for artificial gravity.
NASA and other countries are spending billions of dollars every year on the ISS. But nobody can find a few million dollars to try out one or other of these experiments. Which could easily have been done in the late 1960s or 1970s.
I've written a few articles about this on Science20:
And also written about it a few times here on Quora.
and there was a SpaceShow webinar about it.
Joe Carroll has been advocating his experiment at least since the time of the Shuttle (they could have done it with the Shuttle also - just about all manned flights to orbit have a final stage that goes into orbit at least temporarily and can be used as a counterweight for his experiment).
One way or another, with the will to do it, then it could have been done ages ago, both ideas, tether spin or centrifuge experiments. And I have no idea why we haven't done it.
If there was a will, we could do the tethered Soyuz + final stage experiment probably within a few months or a year - we have lots of experience with tethers now for other purposes in space.
And as for the short arm centrifuge - well -what about putting it into the Bigelow inflated habitat due to fly to the ISS as an experiiment in the next year or two? It is just wide enough for a rather small 4 meter or so diameter small arm centrifuge, like a lightweight hammock - and - there isn't going to be much in it except test equipment + humans go to check it out very now and again. Why not have a small centrifuge in it as well?
In both cases, at least to get a few data points to get us started. About human tolerances of various spin motions (starting with a slow spin say half of lunar gravity and working up) and levels of gravity- and effects on the human physiology of these as well. And having done one such experiment, would probably then be easier to do more and build on the results. Once people realize how easy it is to do and how inexpensive also (for a space experiment). Anyone here got the right connections to do anything about it I wonder?