  We study collections of subrings of $H^*(\overline{\mathcal{M}}_{g,n})$ that
are closed under the tautological operations that map cohomology classes on
moduli spaces of smaller dimension to those on moduli spaces of larger
dimension and contain the tautological subrings. Such extensions of
tautological rings are well-suited for inductive arguments and flexible enough
for a wide range of applications. In particular, we confirm predictions of
Chenevier and Lannes for the $\ell$-adic Galois representations and Hodge
structures that appear in $H^k(\overline{\mathcal{M}}_{g,n})$ for $k = 13$,
$14$, and $15$. We also show that $H^4(\overline{\mathcal{M}}_{g,n})$ is
generated by tautological classes for all $g$ and $n$, confirming a prediction
of Arbarello and Cornalba from the 1990s. In order to establish the final bases
cases needed for the inductive proofs of our main results, we use Mukai's
construction of canonically embedded pentagonal curves of genus 7 as linear
sections of an orthogonal Grassmannian and a decomposition of the diagonal to
show that the pure weight cohomology of $\mathcal{M}_{7,n}$ is generated by
algebraic cycle classes, for $n \leq 3$.
