  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.
