Prime consequences [closed]
If $a,b$ are positive numbers and both small of similar magnitude, then $a^b$ is approximately $1$. Thus if we have a power tower of $n$ small numbers, it seems plausible that the result is either approximately 1 or small, depending on the parity of $n$. This would suggest (heuristic ally) that your and similar sequences won’t converge but rather oscillate. Put differently, you may be able to show convergence for the even and odd index subsequences, respectively.