  Palatini $f(R)$ gravity is probably the simplest extension of general
relativity (GR) and the simplest realization of a metric-affine theory. It has
the same number of degrees of freedom as GR and, in vacuum, it is
straightforwardly mapped into GR with a cosmological constant. The mapping
between GR and Palatini $f(R)$ inside matter is possible but at the expense of
reinterpreting the meaning of the matter fields. The physical meaning and
consequences of such mapping will depend on the physical context. Here we
consider three such cases within the weak field limit: Solar System dynamics,
planetary internal dynamics (seismology), and galaxies. After revising our
previous results on the Solar System and Earth's seismology, we consider here
the possibility of $f(R)$ Palatini as a dark matter candidate. For any $f(R)$
that admits a polynomial approximation in the weak field limit, we show here,
using SPARC data and a recent method that we proposed, that the theory cannot
be used to replace dark matter in galaxies. We also show that the same result
applies to the Eddington-inspired Born-Infeld gravity. Differently from the
metric $f(R)$ case, the rotation curve data are sufficient for this conclusion.
This result does not exclude a combination of modified gravity and dark matter.
