Yes. There's even a practical way to check if a finite deterministic machine has repeated a previous state. Run two of them in parallel, in lockstep so that one takes two steps while the other takes one. If they ever have identical state, they're in a loop. This takes 1.5x the compute power plus 2x as much memory, so it's far cheaper than storing all previous states.

(Some early mainframe era educational interpreter supposedly did this, to stop simple student infinite loops quickly in a batch system.)

Here's a paper on the busy beaver problem which discusses upper bounds.[1] The key here is that the upper bound on states is not computable. However, if you force an upper bound by limiting memory, it becomes computable. But really hard. That paper discusses how very rapidly the number of states grows for that problem.

[1] https://homepages.hass.rpi.edu/heuveb/Teaching/Logic/CompLog...