  In a graph $G$, a vertex dominates itself and its neighbors. A subset
$S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates
every vertex of $G$ at least twice. The double domination number
$\gamma_{\times 2}(G)$ is the minimum cardinality of a double dominating set of
$G$. We show that if $G$ is a maximal outerplanar graph on $n\geq 3$ vertices,
then $\gamma_{\times 2}(G)\leq \lfloor \frac{2n}{3}\rfloor$. Further, if $n\geq
4$, then $\gamma_{\times 2}(G)\leq \min \{\lfloor \frac{n+t}{2}\rfloor, n-t\}$,
where $t$ is the number of vertices of degree $2$ in $G$. These bounds are
shown to be tight. In addition, we also study the case that $G$ is a striped
maximal outerplanar graph.
