  Binary neutron star mergers, which can lead to less massive black holes
relative to other known astrophysical channels, have the potential to probe
modifications to general relativity that arise at smaller curvature scales
compared to more massive compact object binaries. As a representative example
of this, here we study binary neutron star mergers in shift-symmetric
Einstein-scalar-Gauss-Bonnet gravity using evolutions of the full,
nonperturbative evolution equations. We find that the impact on the inspiral is
small, even at large values of the modified gravity coupling (as expected, as
neutron stars do not have scalar charge in this theory). However, postmerger
there can be strong scalar effects, including radiation. When a black hole
forms, it develops scalar charge, impacting the ringdown gravitational wave
signal. In cases where a longer-lived remnant star persists postmerger, we find
that the oscillations of the star source levels of scalar radiation similar to
the black hole formation cases. In remnant stars, we further find that at
coupling values comparable to the maximum value for which black hole solutions
of the same mass exist, there is significant nonlinear enhancement in the
scalar field, which if sufficiently large leads to a breakdown in the
evolution, seemingly due to loss of hyperbolicity of the underlying equations.
