There remain undecidable problems even with finite memory/state space.

Linear bounded automata (LBA) the halting problem is decidable. But many properties of LBA are undecidable:

Emptiness: Does an LBA reject all possible inputs? Universality: Does an LBA accept all possible inputs over its alphabet? Equivalent: Do two LBA accept the same language? Finiteness: Does an LBA accept a finite number of strings.

Computation is the study of infinity. That is how I like to think about it. It doesn't seem that way when you're building a website (well, in some ways because it's not at that point), but every algorithm, data structure, etc is an investigation into a certain part of infinity. Think of the way in which we generally categorize algorithms (Big-O notation)... that's just characterizing infinity.