  We generalize the ultraviolet sector of gravitation via a Born-Infeld action
using lessons from massive gravity. The theory contains all of the elementary
symmetric polynomials and is treated in the Palatini formalism. We show how the
connection can be solved algebraically to be the Levi-Civita connection of an
effective metric. The non-linearity of the algebraic equations yields several
branches, one of which always reduces to General Relativity at low curvatures.
We explore in detail a {\it minimal} version of the theory, for which we study
solutions in the presence of a perfect fluid with special attention to the
cosmological evolution. In vacuum we recover Ricci-flat solutions, but also an
additional physical solution corresponding to an Einstein space. The existence
of two physical branches remains for non-vacuum solutions and, in addition, the
branch that connects to the Einstein space in vacuum is not very sensitive to
the specific value of the energy density. For the branch that connects to the
General Relativity limit we generically find three behaviours for the Hubble
function depending on the equation of state of the fluid, namely: either there
is a maximum value for the energy density that connects continuously with
vacuum, or the energy density can be arbitrarily large but the Hubble function
saturates and remains constant at high energy densities, or the energy density
is unbounded and the Hubble function grows faster than in General Relativity.
The second case is particularly interesting because it could offer an
interesting inflationary epoch even in the presence of a dust component.
Finally, we discuss the possibility of avoiding certain types of singularities
within the minimal model.
