The Frattini subgroup $\Phi(G)$ of a group $G$ is the intersection of all maximal subgroups of $G$; if $G$ has no maximal subgroups, $\Phi(G)=G$ by definition. Frattini subgroups of groups with ``hyperbolic-like” geometry are often small in a suitable sense. Generalizing several known results, we prove that for any countable group $G$ admitting a general type action on a hyperbolic space $S$, the induced action of the Frattini subgroup $\Phi(G)$ on $S$ has bounded orbits, in particular, $\Phi(G)$ has infinite index in $G$. In contrast, we show that the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. The talk is based on a joint work with Gil Goffer and Denis Osin.