I was thinking about generalizing $\uparrow^n$ past the integers so the first problem that came to my mind was what would be $\uparrow^{\frac{1}{2}}$. Firstly, that would be the operation between multiplication and exponentiation, but I couldn't figure out what to do with this information so I wish someone could help me out.
Some tips that can possibly help on the logic are that if $\uparrow^{\frac{1}{2}-n}$, with $n\inℕ$, is discovered it's easy to get $\uparrow^{\frac{1}{2}}$ back, because you can just do $\uparrow^n(\uparrow^{\frac{1}{2}-n})$. That way you can think about the operation between counting and addition and between addition and multiplication and discovering certain properties of any of these can also reveal properties of the proposed problem $\uparrow^{\frac{1}{2}}$.
And lastly, i'm sorry for this question being poorly designed. It's my first question and i'm gonna be better in the next ones.
