  In this note we give three counter-examples which show that the Maximum
Principle generally fails for classical solutions of a system and a single
equation related to the $\infty$-Laplacian. The first is the tangential part of
the $\infty$-Laplace system and the second is the scalar $\infty$-Laplace
equation perturbed by a linear gradient term. The interpretations of the
Maximum Principle for the system are that of the Convex Hull Property and also
of the Maximum Principle of the modulus of the solution.
