  We study inflation with the most general non-degenerate gravitational action
that depends on the symmetric part of the Ricci tensor coupled to a scalar
field in the Palatini formulation of gravity. We use field redefinitions to
shift the effect of the Ricci terms from gravity to the scalar field, and apply
the result to slow-roll inflation.
  As examples, we consider actions quadratic and cubic in the Ricci tensor. In
the quadratic case the results are similar to the case $R+\alpha R^2$ that has
been studied earlier: the tensor-to-scalar ratio $r$ can be suppressed by an
arbitrary amount, while the scalar spectrum is unaffected. In the cubic case,
$r$ can be suppressed by at most a factor of $2/9$, and the change in the
scalar spectral index $n_s$ can be large.
