  An outstanding prediction of general relativity is the fact that the angular
momentum $S$ of an isolated black hole with mass $\mu$ is limited by the Kerr
bound, $S\leq G\mu^2/c$. Testing this cornerstone is challenging due to the
difficulty in modelling spinning compact objects that violate this bound. We
argue that precise, model-independent tests can be achieved by measuring
gravitational waves from an extreme mass ratio inspiral around a supermassive
object, one of the main targets of the future LISA mission. In the extreme mass
ratio limit, the dynamics of the small compact object depends only on its
multipole moments, which are free parameters. At variance with the
comparable-mass case, accurate waveforms are valid also when the spin of the
small object greatly exceeds the Kerr bound. By computing the orbital dephasing
and the gravitational-wave signal emitted by a spinning point particle in
circular, nonprecessing, equatorial motion around a Kerr black hole, we
estimate that LISA will be able to measure the spin of the small compact object
at the level of $10\%$. Together with mass measurements, this will allow for
theory-agnostic, unprecedented constraints on string-theory inspired objects
such as "superspinars", almost in their entire parameter space.
