Will the Universe end?

Warning: you're not going to win or lose any points on this one.

The question of whether the world will end is a perennial one, with The End sometimes forecast to come within a human lifetime or two. This is an ultra-important question, but not the biggest possible one: we can widen our question to whether "The Universe" will end.

Even posing this question is not very straightforward, as "The Universe" has come to mean a great variety of things, from the observable universe that we see through telescopes, all the way through various types of multiverses. So let's start with some definitions.

When we view a particular epoch of the universe through electromagnetic (and now gravitational!) radiation, we are seeing a two-dimensional sphere that we can think of as the "sky" at some "distance." Assembling these nested spheres back to around the nucleosynthesis era era gives a ball of about 46.5 billion light years in radius. This "observiball" can also be thought of as a past "lightcone," and this lightcone and its interior constitute a 3+1 dimensional spacetime region containing every post-nucleosynthesis event that occurred to our past from which information can have reached us traveling at the speed of light or less.

Turning this around to look into the future, we can consider the "Affectiball," or future lightcone, which bounds the region of spacetime that we, here and now on Earth, could reach with sub-lightspeed travel or signaling. Assuming that no future technology allows faster-than-light information transfer (or at least does so only within regions of spacetime pre-engineered for this purposed), everything humanity will ever do or cause will sit within this Affectiball.

We can now ask whether this Affectiball (and its interior) goes on forever, or ends. This really contains two questions. First, will the spacetime go on forever, or terminate in a singularity like the big crunch? Second, will interesting things continue to happen forever, or will the Affectiball approach some sort of equilibrium "heat death"? If we assert that "interesting things" require the ability to do computations, and that computations can't happen without spacetime (see here and here for some discussion), we can combine these into one question:

Is the number of computations that can in principle be done within the Affectiball finite?

If so, we can say the universe will end, at least in terms of anything we can affect or take part in. (The question of whether interesting things will continue to happen elsewhere is an interesting but separate one.)