I make the first estimates of maximum elastic quadrupole deformations sustainable by alternatives to conventional neutron stars. Solid strange quark stars might sustain maximum ellipticities (dimensionless quadrupoles) up to a few times 10-4 rather than a few times 10-7 for conventional neutron stars, and hybrid quark-baryon or meson-condensate stars might sustain up to 10-5. Most of the difference is due to the shear modulus, which can be up to 1033erg/cm3 rather than 1030erg/cm3 in the inner crust of a conventional neutron star. Maximum solid strange star ellipticities are comparable to upper limits obtained for several known pulsars in a recent gravitational-wave search by LIGO. Maximum ellipticities of the more robust hybrid model will be detectable by LIGO at initial design sensitivity. A large shear modulus also strengthens the case for starquakes as an explanation for frequent pulsar glitches.