Puzzler's Edge

Oh, with regards to A(A(x))=a^x (which I'm calling the square hyperroot), I put in the simplest case of y=x^x to Wolfram Alpha, told it to solve for x, and got this mess. So even in the simplest possible situation, I'm not sure fractional tetration makes any sense at all. Especially since, as far as I understand it, the Lambert W-function seems to have multiple outputs for part of its domain.

(although it does somehow give the right answer for e^e=x^x. hmm.)