  The spin angular momentum $S$ of an isolated Kerr black hole is bounded by
the surface area $A$ of its apparent horizon: $8\pi S \le A$, with equality for
extremal black holes. In this paper, we explore the extremality of individual
and common apparent horizons for merging, rapidly spinning binary black holes.
We consider simulations of merging black holes with equal masses $M$ and
initial spin angular momenta aligned with the orbital angular momentum,
including new simulations with spin magnitudes up to $S/M^2 = 0.994$. We
measure the area and (using approximate Killing vectors) the spin on the
individual and common apparent horizons, finding that the inequality $8\pi S <
A$ is satisfied in all cases but is very close to equality on the common
apparent horizon at the instant it first appears. We also introduce a
gauge-invariant lower bound on the extremality by computing the smallest value
that Booth and Fairhurst's extremality parameter can take for any scaling.
Using this lower bound, we conclude that the common horizons are at least
moderately close to extremal just after they appear. Finally, following
Lovelace et al. (2008), we construct quasiequilibrium binary-black-hole initial
data with "overspun" marginally trapped surfaces with $8\pi S > A$ and for
which our lower bound on their Booth-Fairhurst extremality exceeds unity. These
superextremal surfaces are always surrounded by marginally outer trapped
surfaces (i.e., by apparent horizons) with $8\pi S<A$. The extremality lower
bound on the enclosing apparent horizon is always less than unity but can
exceed the value for an extremal Kerr black hole. (Abstract abbreviated.)
