PSR J0537-6910, also known as the Big Glitcher, is the most prolific glitching pulsar known, and its spin-induced pulsations are only detectable in X-ray. We present results from analysis of 2.7 years of NICER timing observations, from 2017 August to 2020 April. We obtain a rotation phase-connected timing model for the entire timespan, which overlaps with the third observing run of LIGO/Virgo, thus enabling the most sensitive gravitational wave searches of this potentially strong gravitational wave-emitting pulsar. We find that the short-term braking index between glitches decreases towards a value of 7 or lower at longer times since the preceding glitch. By combining NICER and RXTE data, we measure a long-term braking index n=-1.25+/-0.01. Our analysis reveals 8 new glitches, the first detected since 2011, near the end of RXTE, with a total NICER and RXTE glitch activity of 8.88x10^-7 yr^-1. The new glitches follow the seemingly unique time-to-next-glitch---glitch-size correlation established previously using RXTE data, with a slope of 5 d microHz^-1. For one glitch around which NICER observes two days on either side, we search for but do not see clear evidence of spectral nor pulse profile changes that may be associated with the glitch.
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