  We study small hairy black holes in a consistent truncation of ${\cal N}=8$
gauged supergravity that consists of a single charged scalar field interacting
with the metric and a U(1) gauge field. Small very near extremal RNAdS black
holes in this system are unstable to decay by superradiant emission. The end
point of this instability is a small hairy black hole that we construct
analytically in a perturbative expansion in its charge. Unlike their RNAdS
counterparts, hairy black hole solutions exist all the way down to the BPS
bound, demonstrating that ${\cal N}=4$ Yang Mills theory has an ${\cal O}(N^2)$
entropy at all energies above supersymmetry. At the BPS bound these black holes
reduce to previously discussed regular, supersymmetric horizon free solitons.
We use numerical methods to continue the construction of these solitons to
large charges and find that the line of soliton solutions terminates at a
singular solution $S$ at a finite charge. We conjecture that a one parameter
family of singular supersymmetric solutions, which emerges out of $S$,
constitutes the BPS limit of hairy black holes at larger values of the charge.
At leading order in their charge, the thermodynamics of the small hairy black
holes constructed in this paper turns out to be correctly reproduced by
modeling these objects as a non interacting mix of an RNAdS black hole and the
supersymmetric soliton in thermal equilibrium. Assuming that a similar non
interacting model continues to apply upon turning on angular momentum, we also
predict a rich family of rotating hairy black holes, including new hairy
supersymmetric black holes. This analysis suggests interesting structure for
the space of (yet to be constructed) hairy charged rotating black holes in
$AdS_5\times S^5$, particularly in the near BPS limit.
