  We consider overdetermined boundary value problems for the $\infty$-Laplacian
in a domain $\Omega$ of $\R^n$ and discuss what kind of implications on the
geometry of $\Omega$ the existence of a solution may have. The classical
$\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the
limit of the $p$-Laplacian as $p\to \infty$ are considered and provide
different answers.
