I don’t know if tetrate is a real word, but tetration is.
Also known as hyper-4 (because it’s the fourth hyperoperation, the first being addition, the second being multiplication, and the third being exponentiation) it might be more familiar to you when I say ‘power tower‘.

Let’s look at an example of a power tower or a tetration.

How do you calculate is? The tetration is not associative, so

The latter evaluates to

but the former (power tower) has to be caculated (by definition or agreement if you like) from above to below or right to left. So it’s

There are a few ways to describe a tetration, like

or

There are more variants of the power tower, like

which we call iterated exponentials. You can calculate it yourself, and it will end up being (remember: from top to bottom!).

A bonus is that tetration and fractals are related. If for with in the complex plane and approaching infinity the tetration will exhibit fractal behavior. No surprise given the repeating behavior of a power tower.