S3 http://pages.bangor.ac.uk/%7Emas010/fig10-6.jpg
I like the Cayley graph of the symmetric group $S_3$ for the presentation $P$ with generators $x,y$ and relations $r=x^3,s=y^2,t=xyxy$. This gives a $2$-complex $K(P)$ with one $0$-cell, two $1$-cells, labelled $x,y$, and three $2$-cells labelled $r,s,t$. The Cayley graph of $P$, i.e. the $1$-skeleton of the universal cover of $K(P)$, is shown above (part of Fig 10.6 of Topology and Groupoids), where the solid lines are mapped to $x$, and the broken lines to $y$. Also shown is a choice of maximal tree. You can also see the lifts of the $2$-cells, but they can't all really be drawn since the corresponding universal cover is a $6$-fold cover, and there are not enough $2$-cells in the above diagram. The dots show where a relation starts.
Note that in group theory, a so called Schreier transversal for a subgroup $H$ of a free group $F$ with generating set $X$ can be seen as a maximal tree in the covering graph of the $1$-complex $K$ determined by the generating set $X$ of $F$, the covering being determined by the subgroup $H$ of $F$.