We investigate the effect of rotation on the maximum mass-loss rate due to an optically-thin radiatively-driven wind according to a formalism which takes into account the possible presence of any instability at the base of the wind that might increase the mass-loss rate. We include the Von Zeipel effect and the oblateness of the star in our calculations. We determine the maximum surface-integrated mass that can be lost from a star by line driving as a function of rotation for a number of relevant stellar models of massive OB stars with luminosities in the range of 5.0 < log (L/L-&ODOT;) < 6.0. We also determine the corresponding maximum loss of angular momentum. We find that rotation increases the maximum mass-loss rate by a moderate factor for stars far from the Eddington limit as long as the ratio of equatorial to critical velocity remains below 0.7. For higher ratios, however, the temperature, flux and Eddington factor distributions change considerably over the stellar surface such that extreme mass loss is induced. Stars close to the Eddington-Gamma limit suffer extreme mass loss already for a low equatorial rotation velocity. We compare the maximum mass-loss rates as a function of rotation velocity with other predicted relations available in the literature which do not take into account possible instabilities at the stellar surface and we find that the inclusion thereof leads to extreme mass loss at much lower rotation rates. We present a scaling law to predict maximum mass-loss rates. Finally, we provide a mass-loss model for the LBVeta Carinae that is able to explain the large observed current mass-loss rate of similar to10(-3) M-circle dot yr(-1) but that leads to too low wind velocities compared to those derived from observations.