Static, spherically symmetric black holes can carry scalar hair when coupling standard Einstein gravity minimally to a self-interacting complex scalar field and a U(1) gauge field. For this scalar hair to exist, the frequency of the scalar field needs to be fine-tuned. In this talk, I will discuss these solutions and point out that for sufficiently large gravitational coupling, the space-time splits into two distinct parts: (a) an inflating interior and (b) an exterior which is described by the extremal Reissner-Nordström solution. Moreover, for a specific range of parameters, the scalar hair develops spatial oscillations, i.e. black holes can carry so-called wavy scalar hair. I will also comment on the globally regular counterparts of these solutions: charged boson stars.