  We investigate the dynamics of a family of functional extensions of the
(Eddington-inspired) Born-Infeld gravity theory, constructed with the inverse
of the metric and the Ricci tensor. We provide a generic formal solution for
the connection and an Einstein-like representation for the metric field
equations of this family of theories. For particular cases we consider
applications to the early-time cosmology and find that non-singular universes
with a cosmic bounce are very generic and robust solutions.
