  The possibility of mining the rotational energy from black holes has
far--reaching implications. Such energy extraction could occur even for
isolated black holes, if hypothetical ultralight bosonic particles exist in
Nature, leading to a new equilibrium state $-$ a black hole with synchronised
bosonic hair $-$ whose lifetime could exceed the age of the Universe. A natural
question is then: for an isolated black hole and at maximal efficiency, how
large is the energy fraction $\epsilon$ that can be extracted from a Kerr black
hole by the superradiant growth of the dominant mode? In other words, how hairy
can the resulting black hole become? A thermodynamical bound for the total
superradiance efficiency, $\epsilon\lesssim 0.29$ (as a fraction of the initial
black hole mass), has long been known, from the area law. However, numerical
simulations exhibiting the growth of the dominant mode only reached about one
third of this value. We show that if the development of superradiant
instabilities is approximately conservative (as suggest by the numerical
evolutions), this efficiency is limited to $\epsilon\lesssim 0.10$, regardless
of the spin of the bosonic field. This is in agreement with the maximum energy
extraction obtained in numerical simulations for a vector field and predicts
the result of similar simulations with a scalar field, yet to be performed.
