Infinitely decreasing exponents

Question

Consider the following expression: $$x=a^{{\frac{1}{2}}^{{\frac{1}{4}}^{\frac{1}{8}...}}}$$ Where $a$ can be pretty much anything. Normally, such exponents are evaluated from top to bottom. But this one goes infinitely.
So, would that expression even make sense? If yes, can something like that converge to a number?

Answer 1

A simple python script will give you approximately 0.81 for the exponent of $a$.

Interestingly, exponent partials appear to oscillate which would suggest that it will not converge.