  The conformal loop ensemble $\operatorname {CLE}_{\kappa}$ with parameter
$8/3<\kappa<8$ is the canonical conformally invariant measure on countably
infinite collections of noncrossing loops in a simply connected domain. Given
$\kappa$ and $\nu$, we compute the almost-sure Hausdorff dimension of the set
of points $z$ for which the number of CLE loops surrounding the disk of radius
$\varepsilon$ centered at $z$ has asymptotic growth $\nu\log (1/\varepsilon )$
as $\varepsilon \to0$. By extending these results to a setting in which the
loops are given i.i.d. weights, we give a CLE-based treatment of the extremes
of the Gaussian free field.
