In this paper, we discuss the quest for more and more education and its implications for social mobility. We document very rapid educational upgrading in Britain over the last 30 years or so and show that this rise has featured faster increases in education acquisition by people from relatively rich family backgrounds. At the same time, wage differentials for the more educated have risen. Putting these two together (more education for people from richer backgrounds and an increase in the pay-off to this education) implies increasing within-generation inequality. By reinforcing already-existing inequalities from the previous generation, this has hindered social mobility. We also highlight three important aspects that, to date, have not been well integrated into the social mobility literature: the acquisition of postgraduate qualifications; gender differences; and the poor education performance of men at the lower end of the education distribution.