  We show that there is a bound depending only on g and [K:Q] for the number of
K-rational points on a hyperelliptic curve C of genus g over a number field K
such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an
explicit bound is 8 r g + 33 (g - 1) + 1.
  The proof is based on Chabauty's method; the new ingredient is an estimate
for the number of zeros of a logarithm in a p-adic `annulus' on the curve,
which generalizes the standard bound on disks. The key observation is that for
a p-adic field k, the set of k-points on C can be covered by a collection of
disks and annuli whose number is bounded in terms of g (and k).
  We also show, strengthening a recent result by Poonen and the author, that
the lower density of hyperelliptic curves of odd degree over Q whose only
rational point is the point at infinity tends to 1 uniformly over families
defined by congruence conditions, as the genus g tends to infinity.
