Addition can be (informally) defined as the application of successor function $S$ on $a$ $b$ times, i.e. $a+b=S\stackrel{b}{\cdots}S a$. Multiplication can be defined as the addition of $a$ with itself $b$ times, i.e. $a\times b=a+\stackrel{b}{\cdots}+a$. Exponentiation can be defined as the multiplication of $a$ with itself $b$ times, i.e. $a\wedge b=a\times\stackrel{b}{\cdots}\times a$.
Intuitively, someone can go one step through this succession and define a function that applies exponentiation of $a$ with itself $b$ times, i.e. $a\star b=a\wedge\stackrel{b}{\cdots}\wedge a$, and repeat this step again and again obtaining an infinite series of functions.
Does this series have a name (at least its fourth element, the next following exponentiation)?
