  A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for
which the first half of the path is assigned the same sequence of colours as
the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is
the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring.
Whether planar graphs have bounded nonrepetitive chromatic number is one of the
most important open problems in the field. Despite this, the best known upper
bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$
upper bound.
