Can we extend our arithmetic beyond the power?

Question

In elementary school we learn the meaning of the following algebraic operations on $\mathbb{C}$: $x+y,xy,x^y$.(sum, product, power) But after that, we do not "go beyond" this operations in the sense I describe below.

Since product is a kind of repeated summation, and power is essentially repeated multiplication, I feel that we should have a notation for $x^{x^{x^{x...}}}$ where there are $n$ occurrences of $x$.

More generally, I want to define a notation $[n]$ recursively(For $a,b \in \mathbb{N}$):

$$ a[1]b=a+b\\ a[n+1]b=a[n]a[n]a[n]...[n]a $$ where there is $b$ occurrences of $a$.

Question: Does such a notation already exists? Can we extend the definition so that a and b can be complex numbers? Can $n$ be non-integral values if we extend this definition?

Answer 1

In fact, there is an operation called "tetration". We have

$$a\uparrow \uparrow b = a\uparrow a\uparrow \cdots \uparrow a\uparrow a$$ with $b$ $a's$ at the right side. This is a power tower of $b$ $a's$ which has to be calculated from above creating huge numbers. Usually, $b$ is a positive integer (I heard of a generalization to real $b's$, but I do not know the details) , and $a$ is usually a positive real number (in most cases, a positive integer).