The Ackermann function is the simplest example of a well-defined total function which is computable but not primitive recursive. See the article "A function to end all functions" by Günter Dotzel, Algorithm 2.4, Oct 1991, Pg 16. The function f(x) = A(x, x) grows much faster than polynomials or exponentials. The definition is:
1. If x = 0 then A(x, y) = y + 1
2. If y = 0 then A(x, y) = A(x-1, 1)
3. Otherwise, A(x, y) = A(x-1, A(x, y-1))