The young, fast-spinning X-ray pulsar J0537-6910 displays an extreme glitch activity, with large spin-ups interrupting its decelerating rotation every ~100 d. We present nearly 13 yr of timing data from this pulsar, obtained with the Rossi X-ray Timing Explorer. We discovered 22 new glitches and performed a consistent analysis of all 45 glitches detected in the complete data span. Our results corroborate the previously reported strong correlation between glitch spin-up size and the time to the next glitch, a relation that has not been observed so far in any other pulsar. The spin evolution is dominated by the glitches, which occur at a rate of ~3.5 per year, and the post-glitch recoveries, which prevail the entire interglitch intervals. This distinctive behaviour provides invaluable insights into the physics of glitches. The observations can be explained with a multicomponent model that accounts for the dynamics of the neutron superfluid present in the crust and core of neutron stars. We place limits on the moment of inertia of the component responsible for the spin-up and, ignoring differential rotation, the velocity difference it can sustain with the crust. Contrary to its rapid decrease between glitches, the spin-down rate increased over the 13 yr, and we find the long-term braking index nl = -1.22(4), the only negative braking index seen in a young pulsar. We briefly discuss the plausible interpretations of this result, which is in stark contrast to the predictions of standard models of pulsar spin-down.
