  We develop a completely bounded counterpart to the non-commutative Choquet
boundary of an operator space. We show how the class of completely bounded
linear maps is too large to accommodate our purposes. To overcome this
obstacle, we isolate the subset of completely bounded linear maps on an
operator space admitting a dilation of the same norm which is multiplicative on
the generated $C^*$-algebra. We view such maps as analogues of the familiar
unital completely contractive maps, and we exhibit many of their structural
properties. Of particular interest to us are those maps which are extremal with
respect to a natural dilation order. We establish the existence of extremals
and show that they have a certain unique extension property. In particular,
they give rise to $*$-homomorphisms which we use to associate to any
representation of an operator space an entire scale of $C^*$-envelopes. We
conjecture that these $C^*$-envelopes are all $*$-isomorphic, and verify this
in some important cases.
