  We study single field slow-roll inflation in the presence of $F(R)$ gravity
in the Palatini formulation. In contrast to metric $F(R)$, when rewritten in
terms of an auxiliary field and moved to the Einstein frame, Palatini $F(R)$
does not develop a new dynamical degree of freedom. However, it is not possible
to solve analytically the constraint equation of the auxiliary field for a
general $F(R)$. We propose a method that allows us to circumvent this issue and
compute the inflationary observables. We apply this method to test scenarios of
the form $F(R) = R + \alpha R^n$ and find that, as in the previously known
$n=2$ case, a large $\alpha$ suppresses the tensor-to-scalar ratio $r$. We also
find that models with $F(R)$ increasing faster than $R^2$ for large $R$ suffer
from numerous problems.
