  The intriguing choice to treat alternative theories of gravity by means of
the Palatini approach, namely elevating the affine connection to the role of
independent variable, contains the seed of some interesting (usually
under-explored) generalizations of General Relativity, the metric-affine
theories of gravity. The peculiar aspect of these theories is to provide a
natural way for matter fields to be coupled to the independent connection
through the covariant derivative built from the connection itself. Adopting a
procedure borrowed from the effective field theory prescriptions, we study the
dynamics of metric-affine theories of increasing order, that in the complete
version include invariants built from curvature, nonmetricity and torsion. We
show that even including terms obtained from nonmetricity and torsion to the
second order density Lagrangian, the connection lacks dynamics and acts as an
auxiliary field that can be algebraically eliminated, resulting in some extra
interactions between metric and matter fields.
