  In a toy model of gauge and gravitational interactions in $D \ge 4$
dimensions, endowed with an invariant UV cut-off $\Lambda$, and containing a
large number $N$ of non-self-interacting matter species, the physical gauge and
gravitational couplings at the cut-off, $\alpha_g \equiv g^2 \Lambda^{D-4}$ and
$\alpha_G \equiv G_N \Lambda^{D-2}$, are shown to be bounded by appropriate
powers of ${1\over N}$. This implies that the infinite-bare-coupling (so-called
compositeness) limit of these theories is smooth, and can even resemble our
world. We argue that such a result, when extended to more realistic situations,
can help avoid large-N violations of entropy bounds, solve the dilaton
stabilization and GUT-scale problems in superstring theory, and provide a new
possible candidate for quintessence.
