  Let F : P^n --> P^n be a morphism of degree d > 1 defined over C. The
dynamical Mordell--Lang conjecture says that the intersection of an orbit
O_F(P) and a subvariety X of P^n is usually finite. We consider the number of
linear subvarieties L in P^n such that the intersection of O_F(P)and L is
"larger than expected." When F is the d'th-power map and the coordinates of P
are multiplicatively independent, we prove that there are only finitely many
linear subvarieties that are "super-spanned" by O_F(P), and further that the
number of such subvarieties is bounded by a function of n, independent of the
point P or the degree d. More generally, we show that there exists a finite
subset S, whose cardinality is bounded in terms of n, such that any n+1 points
in O_F(P)-S are in linear general position in P^n.
