• > Families of 'gigantic finite number' functions found thus far
• > Giant finite number functions that could be represented with ludicrous brain-size/resources

These strike me as basically the same thing relative to my imagination. The biggest numbers mathematicians can describe using the fast-growing hierarchy for the largest computable ordinals are already too gigantic to... well... they're already too gigantic. Taking the Ackermann function as primitive, I still can't visualize the Goodstein sequence of 16, never mind 17, and I think that's somewhere around w^(w^2) in the fast-growing hierarchy.

The jump to uncomputable numbers / numbers that are unique models of second-order axioms would still be a large further jump, though.