  Gravitational-wave astronomy can give us access to the structure of black
holes, potentially probing microscopic or even Planckian corrections at the
horizon scale, as those predicted by some quantum-gravity models of exotic
compact objects. A generic feature of these models is the replacement of the
horizon by a reflective surface. Objects with these properties are prone to the
so-called ergoregion instability when they spin sufficiently fast. We
investigate in detail a simple model consisting of scalar perturbations of a
Kerr geometry with a reflective surface near the horizon. The instability
depends on the spin, on the compactness, and on the reflectivity at the
surface. The instability time scale increases only logarithmically in the
black-hole limit and, for a perfectly reflecting object, this is not enough to
prevent the instability from occurring on dynamical time scales. However, we
find that an absorption rate at the surface as small as 0.4% (reflectivity
coefficient as large as $|{\cal R}|^2=0.996$) is sufficient to quench the
instability completely. Our results suggest that exotic compact objects are not
necessarily ruled out by the ergoregion instability.
