  We construct an analytic self-map $\Phi$ of the bidisk ${\mathbb D}^2$ whose
image touches the distinguished boundary, but whose approximation numbers of
the associated composition operator on $H^2 ({\mathbb D}^2)$ are small in the
sense that $\limsup_{n \to \infty} [a_{n^2} (C_\Phi)]^{1 / n} < 1$.
