Polylogarithms have interesting connections in the Theory of Partitions. This is mainly because a class of generating functions for bivariate partition statistics can be approximated by polylogarithms.
Consider for example the class of generating functions with $s>1,$
$\displaystyle \sum_{n=0}^\infty \sum_{m=0}^\infty p(n,m)x^n u^m =\prod_{k=1}^\infty \frac{1}{(1-xu^k)^{s-2}}. $
When $s=2,$ $p(n,m)$ is the number of partitions of $m$ with $n$ parts. When $s=3,$ $p(n,m)$ represents the number of plane partitions of $m$ with trace $n.$
For simplicity assume $x>0,$ as $u\to ^-1,$ $\displaystyle \prod_{n=1}^\infty \frac{1}{(1-xu^n)^{n^{s-2}}} \thicksim\sqrt[ - \frac{1}{\zeta(2-s)}]{1-x}\exp(Li_{s}(x)/(\ln u)^{s}). $
When used in conjunction with other techniques like the Circle Method, this estimate serves as a foundation for many facts about $p(n,m).$
Off the top of my head here are three papers that use this kind of estimate: http://www.math.drexel.edu/~rboyer/papers/partitions_experimental.pdf http://plms.oxfordjournals.org/content/s2-36/1/117.full.pdf http://www.springerlink.com/content/d65348128235x7m1/