Ackermann function
The smallest number "N" that satisfies the strengthened finite Ramsey theorem is a computable function of "n", "m", "k", but grows extremely fast. In particular it is not primitive recursive, but it is also far larger than standard examples of non-primitive recursive functions such as the Ackermann function. Its growth is so large that Peano arithmetic cannot prove it is defined everywhere, although Peano arithmetic easily proves that the Ackermann function is well defined.