You've never seen a textbook present the series form of e^x?

    exp(x) = summation(x^n/n!, n=0, inf)

exp(0) = 1 is perfectly valid but the identity above only holds for x=0 only if we define 0^0 = 1. And we do!

Fair, infinite series are pretty esoteric. How about derivatives. The power rule:

    d/dx x^n = n x^(n-1)

This identity doesn't hold for n = 1 and x = 0 unless 0^0 = 1.

Eh maybe not, programmers don't use calculus that often. But surely statistics!

    (1 + x)^n = summation((n choose k)x^k, k=0, n)

Take x = 0 and n = 0

    (1 + x)^n = (1 + 0)^0 = 1 = 0^0 = x^0 = x^0 + ... + x^n

0^0 is undefined in general but locally we sometimes need to define it.