  We study Liouville quantum gravity (LQG) surfaces whose law has been
reweighted according to nesting statistics for a conformal loop ensemble (CLE)
relative to $n\in \{0,1,2,\dots\}$ marked points $z_1,\dots,z_n$. The idea is
to consider a reweighting by $\prod_{B\subseteq [n]} e^{\sigma_B N_B}$, where
$\sigma_B$ are real constants and $N_B$ is the number of CLE loops surrounding
the points $z_i$ for $i\in B$. This is made precise via an approximation
procedure where as part of the proof we derive strong spatial independence
results for CLE. The reweighting induces logarithmic singularities for the
Liouville field at $z_1,\dots,z_n$ with a magnitude depending explicitly on
$\sigma_1,\dots,\sigma_n$. We define the partition function of the surface and
explain its relationship to Liouville conformal field theory (CFT) correlation
functions and a potential relationship with a CFT for CLE. In the case of
$n\in\{0,1\}$ points we derive an explicit formula for the partition function.
Furthermore, we obtain a recursive formula for the partition functions where we
express the $n$ point partition function in terms of the partition function for
disks with $k<n$ marked points, and we use this to partially determine for
which values of $(\sigma_B\colon B\subseteq [n])$ the partition function is
finite. The recursive formula is derived via a continuum counterpart of the
peeling process on planar maps, which was earlier studied in works of Miller,
Sheffield, and Werner in the setting of $n=0$ marked points. We also find an
explicit formula for the generator of the Markov process describing the LQG
boundary lengths in the continuum peeling process. Via an explicit calculation
for this Markov process for $n=0$ we give a new proof for the law of the
conformal radius of the outermost CLE loop in the unit disk around 0, which was
earlier established by Schramm, Sheffield, and Wilson.
