We present a timing and glitch analysis of the young X-ray pulsar PSR J0537$-$6910, located within the Large Magellanic Cloud, using 13 years of data from the now decommissioned Rossi X-ray Timing Explorer. Rotating with a spin period of 16 ms, PSR J0537$-$6910 is the fastest spinning and most energetic young pulsar known. It also displays the highest glitch activity of any known pulsar. We have found 42 glitches over the data span, corresponding to a glitch rate of 3.2 yr$^{-1}$, with an overall glitch activity rate of $8.8times 10^{-7},$yr$^{-1}$. The high glitch frequency has allowed us to study the glitch behavior in ways that are inaccessible in other pulsars. We observe a strong linear correlation between spin frequency glitch magnitude and wait time to the following glitch. We also find that the post-glitch spin-down recovery is well described by a single two-component model fit to all glitches for which we have adequate input data. This consists of an exponential amplitude $A = (7.6 pm 1.0)times 10^{-14},$s$^{-2}$ and decay timescale $tau = 27_{-6}^{+7},$d, and linear slope $m = (4.1pm 0.4)times 10^{-16},$s$^{-2},$d$^{-1}$. The latter slope corresponds to a second frequency derivative $ddot{ u} = (4.7pm 0.5) times 10^{-22},$s$^{-3}$, from which we find an implied braking index $n=7.4 pm 0.8$. We also present a maximum-likelihood technique for searching for periods in event-time data, which we used to both confirm previously published values and determine rotation frequencies in later observations. We discuss the implied constraints on glitch models from the observed behavior of this system, which we argue cannot be fully explained in the context of existing theories.
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