Very large or small numbers are useful in combinatorics and number theory. It can also happen that they are needed for intermediate steps in an algorithm, even

Very large or small numbers are useful in combinatorics and number theory. It can also happen that they are needed for intermediate steps in an algorithm, even when the inputs and outputs are moderate (the same reason that 15 digits sometimes isn't enough even when you just want 3 digits of output). Granted, you can generally work around exponent range limitations using tricks (scaling values properly or introducing logarithms), but that can obviously be a hassle.