This dissertation investigates the amenability of topological full groups using a property of group actions called extensive amenability. Extensive amenability is a core concept of several amenability results for groups of dynamical origin. We study its properties and present some applications. The main result of the thesis is such an application, 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. On the other hand, if $G$ is a finitely generated not virtually cyclic group, we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.