bignum libraries usually store big numbers as a sequence of digits (e.g. in base 264). Their limitation is related to the memory available. So the largest number you could represent is tied to the limitation of your virtual address space. You can safely assume that a number even as large as 1010000 is representable in bignum. Of course, a googolplex is not representable as a bignum (because it has more bits than the number of particles in the universe).
Another limitation is the complexity of arithmetic operations. But there exist very efficient bignum algorithms.
FWIW, the GMPlib (a C library for bignums) can deal with numbers as long as there is memory for them. However, it is rumored than when
malloc fails, GMPlib is aborting.
I don't know what happens inside Go bignums when a number is too big to be representable (and that limit varies from one machine to the next and could be different from one run to the next). For example, Go's Int.Mul gives a product whose size is the sum of the size of the arguments, and the "out of memory" error is undocumented (but obviously can happen).
When using bignums, prefer iterative algorithms to recursive ones. For example, a naive recursive factorial might overflow the call stack with large enough bignums, so you want to code it iteratively.