Every finite group can be obtained: if $G$ acts linearly on a finite-dimensional complex vector space, in such a way that the complement of the locus $U$ where the action is free has codimension at least $2$, it is easy to see that the fundamental group of $U/G$ is $G$.
[Edit:] In fact, by an infinite-dimensional variant of this construction it should be possible to obtain any profinite group; I'll post this later.