Applications of general arrow notation?

I don't know where to put this, so I put it here cause it is so advanced...

Anyway, for those who knows what arrow notation is, I have some questions, and for those who don't know what it is, here's a brief explanaition:

The arrow notation is used to apply an operation multiple times ( times) at variable with variable . For example, means that operation is applied times at , with itself. So since is defined as multiplication, means power since is multiplicated times with itself.

All this also means that . For example, .

This can go on forever, meaning that can grov from 1 to 2, from 2 to 3 and so on, creating new operators all the time. For example

This can create **very** big numbers if we continue developing the operators. This is an very interesting invention made by Donald Knuth, but what is it good for? I recently saw an application of when reading about power towers.

Cheers! :)