The question is motivated by this one. It turned out (see my comment there) that the coefficients of the Taylor series for $\log\log x$ at $x=e$ have nice combinatorial description from Sloane's encyclopedia (in the encyclopedia, a related but slightly more complicated function is considered). The coefficients are (up to a power of $e$ multiplied by a factorial) permanents of some easily defined matrices. My question is this:

Is there a combinatorial (possibly 3-dimensional) description of coefficients of the Taylor series of $\log\log\log x$ at $e^e$? Same question for $\log\log\log\log x$, etc.