Using the framework of hierarchical hyperbolicity, we introduce a new compactification of the mapping class group which possesses the robust dynamical features of Thurston's compactification of Teichmuller space but is better suited for studying the asymptotics of subgroups. As an application, we define a notion of geometrical finiteness for subgroups of mapping class groups and prove that several well-studied classes of subgroups are geometrically finite. This is joint work with Mark Hagen and Alessandro Sisto.