  Let $\gamma \in (0,2)$, let $h$ be the planar Gaussian free field, and
consider the $\gamma$-Liouville quantum gravity (LQG) metric associated with
$h$. We show that the essential supremum of the Hausdorff dimension of the
boundary of a $\gamma$-LQG metric ball with respect to the Euclidean (resp.
$\gamma$-LQG) metric is $2 - \frac{\gamma}{d_\gamma}\left(\frac{2}{\gamma} +
\frac{\gamma}{2} \right) + \frac{\gamma^2}{2d_\gamma^2}$ (resp. $d_\gamma-1$),
where $d_\gamma$ is the Hausdorff dimension of the whole plane with respect to
the $\gamma$-LQG metric. For $\gamma = \sqrt{8/3}$, in which case
$d_{\sqrt{8/3}}=4$, we get that the essential supremum of Euclidean (resp.
$\sqrt{8/3}$-LQG) dimension of a $\sqrt{8/3}$-LQG ball boundary is $5/4$ (resp.
$3$).
  We also compute the essential suprema of the Euclidean and $\gamma$-LQG
Hausdorff dimensions of the intersection of a $\gamma$-LQG ball boundary with
the set of metric $\alpha$-thick points of the field $h$ for each $\alpha\in
\mathbb R$. Our results show that the set of $\gamma/d_\gamma$-thick points on
the ball boundary has full Euclidean dimension and the set of $\gamma$-thick
points on the ball boundary has full $\gamma$-LQG dimension.
