  Given a completely arbitrary surface, whether or not it has bounded
curvature, or even whether or not it is complete, there exists an
instantaneously complete Ricci flow evolution of that surface that exists for a
specific amount of time [GT11]. In the case that the underlying Riemann surface
supports a hyperbolic metric, this Ricci flow always exists for all time and
converges (after scaling by a factor $\frac1{2t}$) to this hyperbolic metric
[GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that
there exist complete, bounded curvature initial metrics, including those
conformal to a hyperbolic metric, which have subsequent Ricci flows developing
unbounded curvature at certain intermediate times. In particular, when coupled
with the uniqueness from [Top13], we find that any complete Ricci flow starting
with such initial metrics must develop unbounded curvature over some
intermediate time interval, but that nevertheless, the curvature must later
become bounded and the flow must achieve geometrisation as $t\to\infty$, even
though there are other conformal deformations to hyperbolic metrics that do not
involve unbounded curvature.
  Another consequence of our constructions is that while our Ricci flow from
[GT11] must agree initially with the classical flow of Hamilton and Shi in the
special case that the initial surface is complete and of bounded curvature, by
uniqueness, it is now clear that our flow lasts for a longer time interval in
general, with Shi's flow stopping when the curvature blows up, but our flow
continuing strictly beyond in these situations.
  All our constructions of unbounded curvature developing and then disappearing
are in two dimensions. Generalisations to higher dimensions are then immediate.
