  We present the first examples in black hole thermodynamics of multicritical
phase transitions, in which more than three distinct black hole phases merge at
a critical point. Working in the context of non-linear electrodynamics, we
explicitly present examples of black hole quadruple and quintuple points, and
demonstrate how $n$-tuple critical points can be obtained. Our results indicate
that black holes can have multiple phases beyond the three types observed so
far, resembling the behaviour of multicomponent chemical systems. We discuss
the interpretation of our results in the context of the Gibbs Phase Rule.
