  We consider two types of modifications of Born-Infeld gravity in the Palatini
formulation and explore their dynamics in the early universe. One of these
families considers $f(R)$ corrections to the Born-Infeld Lagrangian, which can
be seen as modifications of the dynamics produced by the quantum effects of
matter, while the other consists on different powers of the elementary building
block of the Born-Infeld Lagrangian, which we denote by $|\hat\Omega|$. We find
that the two types of nonsingular solutions that arise in the original
Born-Infeld theory are also present in these extensions, being bouncing
solutions a stable and robust branch. Singular solutions with a period of
approximate de Sitter inflation are found even in universes dominated by
radiation.
