Empirically we seem to be converging on the idea that the expansion of the universe continues forever (see Wikipedia for a summary of the possibilities), but it's not totally slam-dunk yet. If there is a Big Crunch, then that puts a hard limit on the time available.
If - as we currently believe - that doesn't happen, then the universe will cool over time, until it gets too cold (=too short of negentropy) to sustain any given process. A superintelligence would obviously see this coming, and have plenty of time to prepare - we're talking hundreds of trillions of years before star formation ceases. It might be able to switch to lower-power processes to continue in attenuated form, but eventually it'll run out.
This is, of course, assuming our view of physics is basically right and there aren't any exotic possibilities like punching a hole through to a new, younger universe.
Baring unknown physics, it is absolutely slam-dunk known that the universe ends in a big freeze, not a great crunch. Perlmutter et al got the nobel prize in 2011 for discovering that the expansion of the universe is accelerating, due to an unknown effect called for now dark energy. Unless there is some future undiscovered transition, the end of the universe will be cold and lonely, as even the nearest galaxies eventually red shift to infinity.
As most LWers will know, Clippy the Paperclip Maximiser is a superintelligence who wants to tile the universe with paperclips. The LessWrong wiki entry for Paperclip Maximizer says that:
I think that a massively powerful star-faring entity - whether a Friendly AI, a far-future human civilisation, aliens, or whatever - might indeed end up essentially converting huge swathes of matter in to paperclips. Whether a massively powerful star-faring entity is likely to arise is, of course, a separate question. But if it does arise, it could well want to tile the universe with paperclips.
Let me explain.
To travel across the stars and achieve whatever noble goals you might have (assuming they scale up), you are going to want energy. A lot of energy. Where do you get it? Well, at interstellar scales, your only options are nuclear fusion or maybe fission.
Iron has the strongest binding energy of any nucleus. If you have elements lighter than iron, you can release energy through nuclear fusion - sticking atoms together to make bigger ones. If you have elements heavier than iron, you can release energy through nuclear fission - splitting atoms apart to make smaller ones. We can do this now for a handful of elements (mostly selected isotopes of uranium, plutonium and hydrogen) but we don’t know how to do this for most of the others - yet. But it looks thermodynamically possible. So if you are a massively powerful and massively clever galaxy-hopping agent, you can extract maximum energy for your purposes by taking up all the non-ferrous matter you can find and turning it in to iron, getting energy through fusion or fission as appropriate.
You leave behind you a cold, dark trail of iron.
That seems a little grim. If you have any aesthetic sense, you might want to make it prettier, to leave an enduring sign of values beyond mere energy acquisition. With careful engineering, it would take only a tiny, tiny amount of extra effort to leave the iron arranged in to beautiful shapes. Curves are nice. What do you call a lump of iron arranged in to an artfully-twisted shape? I think we could reasonably call it a paperclip.
Over time, the amount of space that you’ve visited and harvested for energy will increase, and the amount of space available for your noble goals - or for anyone else’s - will decrease. Gradually but steadily, you are converting the universe in to artfully-twisted pieces of iron. To an onlooker who doesn’t see or understand your noble goals, you will look a lot like you are a paperclip maximiser. In Eliezer’s terms, your desire to do so is an instrumental value, not a terminal value. But - conditional on my wild speculations about energy sources here being correct - it’s what you’ll do.