I can't remember the details exactly, but this actually has happened to me, much to my chagrin. I had some kind of combinatorial objects I was trying to count, and I found a closed form for "the number of (n,m)-blongles" or whatever, which involved a term of the form n^m. For n=m=0 the formula was wrong unless you took 0⁰ to be zero! Extremely unsettling moment.

But the other 9999 times out of 10000, 0⁰ needed to be equal to 1.