The largest known prime number (as of February 2023) is 2^82,589,933 − 1
Like that, I guess.
What are the limitations you're anticipating here?
If you can compute with the number, you can send whatever shape you keep it in to compute.
OTOH, actual "very large numbers" like Graham's number can't be computed with meaningfully - writing down even one of the _earlier_ terms in that sequence would need more digits than there are Planck volumes in the universe.