Extending tetration to rational heights (a failed attempt)

This post is attempting to answer one of the questions at the end of Let’s build an operator which introduces the idea of hyperoperations. It’s not strictly necessary to read that post first, but I’d recommend it. \[\newcommand{\tetration}[2]{ {}^{#2}#1 }\] Repeated addition turns into multiplication. Repeated multiplication turns into exponentiation. What does repeated exponentiation turn into?…