TITLE: Is there a notation for the repetition of basic operations?
QUESTION [5 upvotes]: I mean... 

multiplication is the repetition of addition:


$2*2 = 2+2$
$3*3 = 3+3+3$

exponential is the repetition of multiplication:


$2^2 = 2 * 2$
$3^3 = 3*3*3$


.. it is an obvious pattern.
I propose:

Addition is rank 1
Multiplication is rank 2 
Exponential is rank 3 
etc ...

This would mean
$H(r, x)$ is the $r$th rank operation on $x$. 
For example: $H(3, 2) = 2^2$ 
and $H(4, 3) = {^{^33}}3$
It is already kind of wire to get this value since it is quite big. But the function of the rank is certainly one that rises fast.
Does anyone know more about this kind of stuff?

REPLY [4 votes]: There is also the notation $a^{n*} $ for $\underbrace {a*\cdots*a}_{\text {$n $ factors} }$. Using this notation we have
$$na=a^{n+} $$
$$a^n=a^{n\cdot} $$