  Viable corrections to the matter sector of Poisson's equation may result in
qualitatively different astrophysical phenomenology, for example the
gravitational collapse and the properties of compact objects can change
drastically. We discuss a class of modified non-relativistic theories and focus
on a relativistic completion, Eddington-inspired Born-Infeld gravity. This
recently proposed theory is equivalent to General Relativity in vacuum, but its
non-trivial coupling to matter prevents singularities in early cosmology and in
the non-relativistic collapse of non-interacting particles. We extend our
previous analysis, discussing further developments. We present a full numerical
study of spherically symmetric non-relativistic gravitational collapse of dust.
For any positive coupling, the final state of the collapse is a regular
pressureless star rather than a singularity. We also argue that there is no
Chandrasekhar limit for the mass of non-relativistic white dwarf in this
theory. Finally, we extend our previous results in the fully relativistic
theory by constructing static and slowly rotating compact stars governed by
nuclear-physics inspired equations of state. In the relativistic theory, there
exists an upper bound on the mass of compact objects, suggesting that black
holes can still be formed in the relativistic collapse.
