  We study the cosmology of the complete quadratic (in torsion and
nonmetricity) metric-affine gravity. Namely, we add to the scalar-curvature
gravitational Lagrangian, the 17 independent quadratic (parity-even and
parity-odd) torsion and nonmetricity invariants. Sticking to a homogeneous and
isotropic Friedmann-Robertson-Walker spacetime and assuming a perfect
hyperfluid source, we explore the new effects that torsion and nonmetricity
bring into play. It is shown that the inclusion of these invariants offers rich
phenomenology. In particular, some well-known examples of exotic matter like
cosmic strings, domain walls, stiff matter, etc., emerge quite naturally as
manifestations of the fluid's intrinsic structure (hypermomentum). By studying
the extended Friedmann equations in the complete quadratic theory and isolating
the various parts of the hypermomentum, we find a plethora of solutions with
interesting features.
