  We consider infinite sequences {R_n} of successive local quadratic transforms
of a regular local ring. Let S denote the directed union of the sequence of
regular local rings R_n. We previously showed the existence of a unique limit
point V of the family of order valuation rings of the sequence. In this paper,
we examine asymptotic properties of this family of order valuations. We link
this asymptotic behavior to ring-theoretic properties of S, namely whether S is
archimedean and whether S is completely integrally closed. We construct
examples of such S that are archimedean and completely integrally closed but
not valuation domains. We give an explicit description of V, where the
description depends on whether S is archimedean or non-archimedean.
