  We compute the N$^3$LO gravitational quadratic-in-spin interactions at $G^4$
in the post-Newtonian (PN) expansion via the effective field theory (EFT) of
gravitating spinning objects for the first time. This result contributes at the
$5$PN order for maximally-spinning compact objects, adding the spinning case to
the static sector at this PN accuracy. This sector requires extending the EFT
of a spinning particle beyond linear order in the curvature to include
higher-order operators quadratic in the curvature that are relevant at this PN
order. We make use of a diagrammatic expansion in the worldline picture, and
rely on our recent upgrade of the \texttt{EFTofPNG} code, which we further
extend to handle this sector. Similar to the spin-orbit sector, we find that
the contributing three-loop graphs give rise to divergences, logarithms, and
transcendental numbers. However, in this sector all of these features conspire
to cancel out from the final result, which contains only finite rational terms.
