What is "superfunction"

In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could be defined by the expression

$$S(z;x) = \underbrace{f\Big(f\big(\dots f(x)\dots\big)\Big)}_{z \text{ evaluations of the function }f} .$$
Then, S(z;x) can be interpreted as the superfunction of the function f(x). Such a definition is valid only for a positive integer index z. The variable x is often omitted. Much study and many applications of superfunctions employ various extensions of these superfunctions to complex and continuous indices; and the analysis of the existence, uniqueness and their evaluation. The Ackermann functions and tetration can be interpreted in terms of super-functions.