  We prove a Tits alternative for topological full groups of minimal actions of
finitely generated groups. On the one hand, we show that topological full
groups of minimal actions of virtually cyclic groups are amenable. By doing so,
we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On
the other hand, when a finitely generated group $G$ is not virtually cyclic,
then we construct a minimal free action of $G$ on a Cantor space such that the
topological full group contains a non-abelian free group.
