Is there an "exponentiation for exponentiation?"

Question

I have asked a few colleagues and none of them knew of one, so I figure Math.SE is probably the best place to ask.

Is there a name for exponentiation repeating $n$ times, like multiplication is for addition and exponentiation is for multiplication?

Answer 1

Tetration. If you want to get into repeated tetration, repeated repeated tetration, and so on, the general term is hyperoperation.

(But be warned: none of these operations are nearly as nice as addition, multiplication, or even exponentiation — for example, there isn't a particularly natural way to define them for non-integer arguments.)