Fun fact: Ackermann φ(m,n,p) function does exactly this. Every new order repeats the operation before. You can think of it as "addition of the order p" and extend arithmetic operations.

Ackermann used this function for a completely different purpose:  the function illustrates that not all total computable functions are primitive recursive.
more: http://en.wikipedia.org/wiki/Ackermann_function

See also http://en.wikipedia.org/wiki/Hyperoperation for more information on the hyper operation sequence. The correct name for repeated tetration is pentation.