Solving for f one finds a ``continued fraction"-like answer
f(x) = (log x)/(log(log x)/(log(log x)/(log ...)))
This question has been repeated here from time to time over the years, and no one seems to have heard of any published work on it, nor a published name for it.
This function is the inverse of f(x) = x^x. It might be argued that such description is good enough as far as mathematical names go: "the inverse of the function f(x) = x^x" seems to be clear and succint.
Another possible name is lx(x). This comes from the fact that the inverse of e^x is ln(x) thus the inverse of x^x could be named lx(x).
It's not an analytic function.
The ``continued fraction" form for its numeric solution is highly unstable in the region of its minimum at 1/e (because the graph is quite flat there yet logarithmic approximation oscillates wildly), although it converges fairly quickly elsewhere. To compute its value near 1/e, use the bisection method which gives good results. Bisection in other regions converges much more slowly than the logarithmic continued fraction form, so a hybrid of the two seems suitable. Note that it's dual valued for the reals (and many valued complex for negative reals).
A similar function is a built-in function in MAPLE called W(x) or Lambert's W function. MAPLE considers a solution in terms of W(x) as a closed form (like the erf function). W is defined as W(x)e^(W(x)) = x.
Notice that f(x) = exp(W(log(x))) is the solution to f(x)^f(x) = x
An extensive treatise on the known facts of Lambert's W function is available for anonymous ftp at dragon.uwaterloo.ca at /cs-archive/CS-93-03/W.ps.Z.