The generalisation that you're hinting at is known as Knuth's up-arrow notation [0].
There's a number known as Graham's number [1] which is defined in terms of up-arrow notation and was for a while the largest specific positive integer to have been used in a mathematical proof.
There's a number known as Graham's number [1] which is defined in terms of up-arrow notation and was for a while the largest specific positive integer to have been used in a mathematical proof.
[0] https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
[1] https://en.wikipedia.org/wiki/Graham%27s_number