  The planar integrability of $\mathcal{N}=4$ super-Yang-Mills (SYM) is the
cornerstone for numerous exact observables. We show that the large charge
sector of the ${\rm SU}(2)$ $\mathcal{N}=4$ SYM provides another interesting
solvable corner which exhibits striking similarities despite being far from the
planar limit. We study non-BPS operators obtained by small deformations of
half-BPS operators with $R$-charge $J$ in the limit $J\to\infty$ with
$\lambda_{J}\equiv g_{\rm YM}^2 J/2$ fixed. The dynamics in this {\it large
charge 't Hooft limit} is constrained by a centrally-extended
$\mathfrak{psu}(2|2)^2$ symmetry that played a crucial role for the planar
integrability. To the leading order in $1/J$, the spectrum is fully fixed by
this symmetry, manifesting the magnon dispersion relation familiar from the
planar limit, while it is constrained up to a few constants at the next order.
We also determine the structure constant of two large charge operators and the
Konishi operator, revealing a rich structure interpolating between the
perturbative series at weak coupling and the worldline instantons at strong
coupling. In addition we compute heavy-heavy-light-light (HHLL) four-point
functions of half-BPS operators in terms of resummed conformal integrals and
recast them into an integral form reminiscent of the hexagon formalism in the
planar limit. For general ${\rm SU}(N)$ gauge groups, we study integrated HHLL
correlators by supersymmetric localization and identify a dual matrix model of
size $J/2$ that reproduces our large charge result at $N=2$. Finally we discuss
a relation to the physics on the Coulomb branch and explain how the dilaton
Ward identity emerges from a limit of the conformal block expansion. We comment
on generalizations including the large spin 't Hooft limit, the combined large
$N$-large $J$ limits, and applications to general $\mathcal{N}=2$
superconformal field theories.
