  There is a significant possibility that astrophysical black holes with
nearly-extremal spins exist. Numerical simulations of such systems require
suitable initial data. In this paper, we examine three methods of constructing
binary-black-hole initial data, focusing on their ability to generate black
holes with nearly-extremal spins: (i) Bowen-York initial data, including
standard puncture data (based on conformal flatness and Bowen-York extrinsic
curvature), (ii) standard quasi-equilibrium initial data (based on the
extended-conformal-thin-sandwich equations, conformal flatness, and maximal
slicing), and (iii) quasi-equilibrium data based on the superposition of
Kerr-Schild metrics. We find that the two conformally-flat methods (i) and (ii)
perform similarly, with spins up to about 0.99 obtainable at the initial time.
However, in an evolution, we expect the spin to quickly relax to a
significantly smaller value around 0.93 as the initial geometry relaxes. For
quasi-equilibrium superposed Kerr-Schild (SKS) data [method (iii)], we
construct initial data with \emph{initial} spins as large as 0.9997. We evolve
SKS data sets with spins of 0.93 and 0.97 and find that the spin drops by only
a few parts in 10^4 during the initial relaxation; therefore, we expect that
SKS initial data will allow evolutions of binary black holes with relaxed spins
above 0.99. [Abstract abbreviated; full abstract also mentions several
secondary results.]
