  Characterizing the properties of black holes is one of the most important
science objectives for gravitational-wave observations. Astrophysical evidence
suggests that black holes that are nearly extremal (i.e. spins near the
theoretical upper limit) might exist and thus might be among the merging black
holes observed with gravitational waves. In this paper, we explore how well
current gravitational wave parameter estimation methods can measure the spins
of rapidly spinning black holes in binaries. We simulate gravitational-wave
signals using numerical-relativity waveforms for nearly-extremal, merging black
holes. For simplicity, we confine our attention to binaries with spins parallel
or antiparallel with the orbital angular momentum. We find that recovering the
holes' nearly extremal spins is challenging. When the spins are nearly extremal
and parallel to each other, the resulting parameter estimates do recover spins
that are large, though the recovered spin magnitudes are still significantly
smaller than the true spin magnitudes. When the spins are nearly extremal and
antiparallel to each other, the resulting parameter estimates recover the small
effective spin but incorrectly estimate the individual spins as nearly zero. We
study the effect of spin priors and argue that a commonly used prior (uniform
in spin magnitude and direction) hinders unbiased recovery of large black-hole
spins.
