The question is motivated by  [this one.](https://mathoverflow.net/questions/57588/series-for-loglog-closed)  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](http://oeis.org/search?q=1%2C-2%2C7%2C-35%2C228%2C+1834&language=english&go=Search)  (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. 



about power series for iterated logarithms
I think it may be simpler to deal with the Maclaurin series for the functions $\log(1-x)$, $-\log(1+\log(1-x))$, $-\log(1+\log(1+\log(1-x)))$, etc. The third one, for example, is the exponential generating function for $1,3,15,105,947,10472,137337,\dots$ which is <http://oeis.org/A000268> and there are a couple of references there which may be worth tracking down, J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353 and P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5 (and a couple of others, besides). 


