  We describe a model that generates first order adiabatic EMRI waveforms for
quasi-circular equatorial inspirals of compact objects into rapidly rotating
(near-extremal) black holes. Using our model, we show that LISA could measure
the spin parameter of near-extremal black holes (for $a \gtrsim 0.9999$) with
extraordinary precision, $\sim$ 3-4 orders of magnitude better than for
moderate spins, $a \sim 0.9$. Such spin measurements would be one of the
tightest measurements of an astrophysical parameter within a gravitational wave
context. Our results are primarily based off a Fisher matrix analysis, but are
verified using both frequentest and Bayesian techniques. We present analytical
arguments that explain these high spin precision measurements. The high
precision arises from the spin dependence of the radial inspiral evolution,
which is dominated by geodesic properties of the secondary orbit, rather than
radiation reaction. High precision measurements are only possible if we observe
the exponential damping of the signal that is characteristic of the
near-horizon regime of near-extremal inspirals. Our results demonstrate that,
if such black holes exist, LISA would be able to successfully identify rapidly
rotating black holes up to $a = 1-10^{-9}$ , far past the Thorne limit of $a =
0.998$.
