  We consider the question of which semigroups can occur as the semigroup
$S_R(\nu)$ of positive values of a rank 1 valuation dominating a Noetherian
local ring $R$. We give a number of bounds of polynomial type on the growth of
$\phi(n)=S_R(\nu)\cap (0,n)$ for $n\in\NN$, starting with the upper bound of
$P_R(n)$, where $P_R(n)$ is the Hilbert function of $R$. This bound is
generalized to an extremely general bound for arbitrary rank valuations in the
paper "Semigroups of valuations on local rings, II", by Cutkosky and Teissier,
arXiv:0805.3788. This bound is already enough to give simple examples of rank 1
well ordered semigroups which are not the value semigroup $S_R(\nu)$ of a
valuation dominating a Noetherian local ring.
  In the case of rank 1, it is possible to give more precise estimates of
$\phi(n)$, which we prove in this paper. We also give examples showing that
many different rates of growth are possible for $\phi(n)$ on a regular local
ring of dimension 2, such as $n({\alpha}$ for any rational $\alpha$ with
$1\le\alpha\le 2$, and $n{log}(n)$.
