Expanding on some of this... I suppose it makes no sense unless $x$ is between 0 and 1. And in this case... Define $y =x^{x^{x^{\cdots}}}$. Then we have $y= x^y$. For any fixed $0<x<1$, this defines a unique positive value for $y$ (intermediate value theorem will imply $y$ is between 0 and 1). So that's one way to define it in the range we want. It seems to me that solving the above equation gives us: $$x = e^{\log(y)/y} = y^{1/y}$$. So your function is the inverse of this. Might be related to the Lambert W function (didn't think).