  We consider the Discrete Gaussian Free Field (DGFF) in domains
$D_N\subseteq\mathbb Z^2$ arising, via scaling by $N$, from nice domains
$D\subseteq\mathbb R^2$. We study the statistics of the values order
$\sqrt{\log N}$ below the absolute maximum. Encoded as a point process on
$D\times\mathbb R$, the scaled spatial distribution of these near-extremal
level sets in $D_N$ and the field values (in units of $\sqrt{\log N}$ below the
absolute maximum) tends, as $N\to\infty$, in law to the product of the critical
Liouville Quantum Gravity (cLQG) $Z^D$ and the Rayleigh law. The convergence
holds jointly with the extremal process, for which $Z^D$ enters as the
intensity measure of the limiting Poisson point process, and that of the DGFF
itself; the cLQG defined by the limit field then coincides with $Z^D$. While
the limit near-extremal process is measurable with respect to the limit
continuum GFF, the limit extremal process is not. Our results explain why the
various ways to "norm" the lattice cLQG measure lead to the same limit object,
modulo overall normalization.
