We develop a theoretical treatment that allows us to determine the maximum mass-loss rate of a hot rotating star with a wind that is accelerated by radiation pressure due to spectral lines, taking into account finite disk correction as well as the effect of photon tiring but neglecting multiple scattering. The maximum mass-loss rate of a star is obtained by subsequent numerical integrations of the momentum equation from an assumed position of the sonic point onwards for increasing values of the mass loss, until the wind can no longer escape. For stars rotating below 80% of the critical velocity the decrease in the velocity far out in the wind due to the maximisation of the mass loss is negligible. Stars rotating at > 80% of the critical speed have a kinked velocity law connected with the highest possible mass-loss rate. In such cases the wind velocity increases up to typically a few stellar radii, and decreases subsequently almost ballistically outwards. In these cases the terminal wind velocity is much smaller than the maximum wind velocity. For O-type main-sequence stars, the maximum mass-loss rates derived from our formalism are somewhat smaller than those derived for self-regulated line-driven winds including multiple scattering. For B-type supergiants, however, the maximum mass-loss rate is higher by about a factor 1.5-2. Including rotation, but without gravity darkening, results in a maximum mass-loss rate that is twice as high as for a non-rotating star.