  We introduce the growth rank of a C*-algebra, a (N \cup {\infty})-valued
invariant which measures how far an algebra is from absorbing the Jiang-Su
algebra Z tensorially. We prove that its range is exhausted by simple nuclear
C*-algebras, and obtain in the process a well developed theory of unbounded
dimension growth for approximately homogeneous (AH) algebras. Another
consequence of the range result is the existence of a simple, nuclear, and
non-Z-stable C*-algebra which is not tensorially prime. The properties of the
growth rank suggest a universal property which may be considered inside any
class of unital and nuclear C*-algebras. We prove that Z satisfies this
property inside a class of locally subhomogeneous algebras.
