How many registers does a computer *need*?
You need to make your question a bit more precise.

First, regarding your request to avoid "unlimited" registers -- I assume you mean registers that can hold an arbitrary value (e.g., any natural number).

If you a-priori bound your registers, then the possible states your machine can be in is finite. That is, you have a deterministic finite automaton (or transducer, if you allow inputs and outputs). In this case, you don't need any register, but the machine would be huge without them. For example, you can replace 32 binary registers with $2^{32}$ states.

If you do allow arbitrary values, then 2 counters suffice to simulate a Turing Machine using a Minsky Counter Machine, under certain assumptions.