  The maximum mass of a nonrotating neutron star, $M_{\rm TOV}$, plays a very
important role in deciphering the structure and composition of neutron stars
and in revealing the equation of state (EOS) of nuclear matter. Although with a
large-error bar, the recent mass estimate for the black-widow binary pulsar PSR
J0952-0607, i.e. $M=2.35\pm0.17~M_\odot$, provides the strongest lower bound on
$M_{\rm TOV}$ and suggests that neutron stars with very large masses can in
principle be observed. Adopting an agnostic modelling of the EOS, we study the
impact that large masses have on the neutron-star properties. In particular, we
show that assuming $M_{\rm TOV}\gtrsim 2.35\,M_\odot$ constrains tightly the
behaviour of the pressure as a function of the energy density and moves the
lower bounds for the stellar radii to values that are significantly larger than
those constrained by the NICER measurements, rendering the latter ineffective
in constraining the EOS. We also provide updated analytic expressions for the
lower bound on the binary tidal deformability in terms of the chirp mass and
show how larger bounds on $M_{\rm TOV}$ lead to tighter constraints for this
quantity. In addition, we point out a novel quasi-universal relation for the
pressure profile inside neutron stars that is only weakly dependent from the
EOS and the maximum-mass constraint. Finally, we study how the sound speed and
the conformal anomaly are distributed inside neutron stars and show how these
quantities depend on the imposed maximum-mass constraints.
