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).