How do we learn about the phases of matter beyond nuclear density? They are to be found only in the interior of neutron stars, which are inaccessible and hard to observe. One approach is through the oscillations of neutron stars, which depend on the viscosity of their interior. If the viscosity is low enough then “r-mode” oscillations arise spontaneously and cause the star to spin down. Finding fast-spinning stars therefore puts limits on the viscosity viscosity and hence on the possible phases present in the interior of the star. This talk discusses non-linear effects which arise for large amplitude “suprathermal” oscillations. This is relevant to unstable modes such as r-modes, which grow in amplitude until saturated by non-linear effects. We find that find that the viscosity typically grows rapidly with amplitude in the suprathermal regime. Our results indicate that suprathermal bulk viscosity may be the limiting factor in the growth of r-modes, determining their saturation amplitude and hence the amount of time required for an r-mode to spin down a neutron star.