Mathematics
Primitive Recursive Functions and Ackermann's Function
Quick fact
Ackermann's function was the first published example of a total computable function that is not primitive recursive; it grows faster than any primitive recursive function, yet it can be computed with a simple set of recursive rules.
Why this is interesting
You know that a computer can compute anything given enough time, but could there be a function that is computable yet impossible to express with simple recursion loops? Ackermann's function is exactly that surprising example.