The Ackermann function is the simplest example of a well-defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive. It grows faster than an exponential function, or even a multiple exponential function.

Ackermann number | computable function | Goodstein sequence | power tower | primitive recursive function | TAK function | total function

Wilhelm Ackermann