The idea behind the notation is quite simple but not extremely useful in real-life situations and it's not mainstream mathematics.
We can think of multiplication as shorthand addition. Adding a number x to itself n times is the same as multplying x by n. Exponentiation (taking powers) is shorthand multiplication in the same way. It's quite natural to think about shorthand exponentiation. It's called tetration. Shorthand tetration is pentation and this is the operation I used to write my number.