  We study single-field slow-roll inflation embedded in Palatini $F(R)$ gravity
where $F(R)$ grows faster than $R^2$. Surprisingly, the consistency of the
theory requires the Jordan frame inflaton potential to be unbounded from below.
Even more surprisingly, this corresponds to an Einstein frame inflaton
potential bounded from below and positive definite. We prove that for all such
Palatini $F(R)$'s, there exists a universal strong coupling limit corresponding
to a quadratic $F(R)$ with the wrong sign for the linear term and a
cosmological constant in the Jordan frame. In such a limit, the
tensor-to-scalar ratio $r$ does not depend on the original inflaton potential,
while the scalar spectral index $n_s$ does. Unfortunately, the system is
ill-defined out of the slow-roll regime. A possible way out is to upgrade to a
$F(R,X)$ model, with $X$ the Jordan frame inflaton kinetic term. Such a
modification essentially leaves the inflationary predictions unaffected.
