Let X, Y be complex spaces,
and f : X ⇁ Y a meromorphic map. Assume in Y an admissible family A= {Sb}b∈N
of analytic subsets Sb is given. Assume f is almost adapted to A. The purpose of this
paper is to prove that, if f satisfies certain growth conditions, the valence of Sb (for
almost all Sb∈ A) grows to infinity at the same rate as the characteristic of f. Here
X is assumed to carry an exhaustion function which is, e.g., g-concave, centrally
g-convex or g-quasiparabolic.