  We study partition functions for the dimer model on families of finite graphs
converging to infinite self-similar graphs and forming approximation sequences
to certain well-known fractals. The graphs that we consider are provided by
actions of finitely generated groups by automorphisms on rooted trees, and thus
their edges are naturally labeled by the generators of the group. It is thus
natural to consider weight functions on these graphs taking different values
according to the labeling. We study in detail the well-known example of the
Hanoi Towers group $H^{(3)}$, closely related to the Sierpi\'nski gasket.
