Not yet reviewed by Pith; the record is open.

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.