I think the best answer is that science does not know enough to answer this question.
Most of physics is done using real numbers with an infinite number of states. However, we do see integers (and thus our best chance at a finite number of states) in quantum mechanics. In particular, the standard model uses them heavily. They appear because the only stable states for some systems appear with integer numbers.
However, when we start talking about all of the universe for all time, it gets murky. If there exists one real numbered variable which is not selected from the integers, then the universe would have to be uncountably infinite.
The best argument I have heard for there being a finite number of states is Plank time and Plank length. These are astronomically short times and lengths, far beyond anything we can measure. However, we do believe that our models do not describe anything shorter or faster than that. So if it turns out there's nothing new to model, then that would suggest that these Plank times and lengths discretize everything. But when in science has there not been something new to model?
And, of course, you mention the big crunch. Other models do make it harder to tell what the answer should be. If you have a slow cold death, the number of states may balloon, but are they infinite? That's a pretty demanding bar.
The difference between an countably infinite number of states and a very large number is very hard to discern. Indeed, it would take great hubris to feel that we can meaningfully distinguish between Graham's Number and infinity. Yes, mathematics does indeed have that huberis, but science has to deal with things differently. (Graham's number is so big that you can't write it down, 1 digit per atom in the universe. In fact, if you try to write down the number of digits, the resulting number is still larger than the number of atoms in the universe... by an amount so large that its hard to convince yourself this log log process even did anything to the number).
All that being said, if I ask some of my friends in other circles, they might point out that there is only one state the universe can be in: the Is. What is is, and nothing else can be. I give that one for some perspective. The question of "number of states of the universe" does indeed imply that there could exist other universes in different states. But that goes metaphysical very quickly. It just seemed like some good food for thought.