  We further develop the string $1/c^2$ expansion of closed bosonic string
theory, where $c$ is the speed of light. The expansion will be performed up to
and including the next-to-next-to-leading order (NNLO). We show that the
next-to-leading order (NLO) theory is equal to the Gomis--Ooguri string,
generalised to a curved target space, provided the target space geometry admits
a certain class of co-dimension-2 foliations. We compute the energy of the
string up to NNLO for a flat target space with a circle that must be wound by
the string, and we show that it agrees with the $1/c^2$ expansion of the
relativistic energy. We also compute the algebra of Noether charges for a flat
target space and show that this matches order-by-order with an appropriate
expansion of the Poincar\'e algebra, which at NLO gives the string Bargmann
algebra. Finally, we expand the phase space action, which allows us to perform
the Dirac procedure and pass to the quantum theory. It turns out that the
Poisson brackets change at each order, and we show that the normal ordering
constant of the relativistic theory, which does not depend on $c$, can be
reproduced by the NLO and NNLO theories.
