  Given any closed, connected, orientable $3$--manifold and integers $g\geq
g(M), D > 0$, we show the existence of knots in $M$ whose genus $g$ bridge
number is greater than $D$. These knots lie in a page of an open book
decomposition of $M$, and the proof proceeds by examining the action of the map
induced by the monodromy on the arc and curve complex of a page. A corollary is
that there are Berge knots of arbitrarily large genus one bridge number.
