What are the minimum memory requirments a microprocessors must have to perform any calculation?
unlimited complexity?

That requires unlimited memory, after all - the numbers must be stored somewhere, and the place to store them is always in memory.

So you need 1 memory 'cell' for each number, and another for each operation. You also need a counter to keep track of the next number or operation. Add to that a register to perform the calculation (only 1 is needed as you can use it to store the results in-place. In reality you'll use another register to notify you of errors such as overflow or division by zero).

Or to put it another way: you need sufficient RAM, an accumulator and a program counter. You can get away without the counter if the memory are streamed in sequentially on a specialist computing machine (eg on tape as per Euphoric's comment).