You can have inverse operations for the higher operations as well. 4^4 is 256, so you can think of 4 as the "tetrated root" of 256. Also see this

(I'm using 'tetrating' as a term for the operation after exponentiation: in other words, 4 tetrated to the 4th is 4^(4^(4^4))).

Two problems: there may not be a clear way to define tetrating and higher operations to fractional amounts, and exponentiation and up aren't associative, so you need a convention for what to do with the parentheses.