  Black holes of mass M must have a spin angular momentum S below the Kerr
limit chi = S/M^2 < 1, but whether astrophysical black holes can attain this
limiting spin depends on their accretion history. Gas accretion from a thin
disk limits the black-hole spin to chi_gas < 0.9980 +- 0.0002, as
electromagnetic radiation from this disk with retrograde angular momentum is
preferentially absorbed by the black hole. Extrapolation of
numerical-relativity simulations of equal-mass binary black-hole mergers to
maximum initial spins suggests these mergers yield a maximum spin chi_eq <
0.95. Here we show that for smaller mass ratios q = m/M << 1, the superradiant
extraction of angular momentum from the larger black hole imposes a fundamental
limit chi_lim < 0.9979 +- 0.0001 on the final black-hole spin even in the
test-particle limit q -> 0 of binary black-hole mergers. The nearly equal
values of chi_gas and chi_lim imply that measurement of supermassive black-hole
spins cannot distinguish a black hole built by gas accretion from one assembled
by the gravitational inspiral of a disk of compact stellar remnants. We also
show how superradiant scattering alters the mass and spin predicted by models
derived from extrapolating test-particle mergers to finite mass ratios.
