  Motivated by the unknown nature of the $2.50-2.67\,M_\odot$ compact object in
the binary merger event GW190814, we study the maximum neutron star mass based
on constraints from low-energy nuclear physics, neutron star tidal
deformabilities from GW170817, and simultaneous mass-radius measurements of PSR
J0030+045 from NICER. Our prior distribution is based on a combination of
nuclear modeling valid in the vicinity of normal nuclear densities together
with the assumption of a maximally stiff equation of state at high densities, a
choice that enables us to probe the connection between observed heavy neutron
stars and the transition density at which conventional nuclear physics models
must break down. We demonstrate that a modification of the highly uncertain
supra-saturation density equation of state beyond 2.64 times normal nuclear
density is required in order for chiral effective field theory models to be
consistent with current neutron star observations and the existence of
$2.6\,M_\odot$ neutron stars. We also show that the existence of very massive
neutron stars strongly impacts the radii of $\sim 2.0\,M_\odot$ neutron stars
(but not necessarily the radii of $1.4\,M_\odot$ neutron stars), which further
motivates future NICER radius measurements of PSR J1614-2230 and PSR
J0740+6620.
