  We take string theory in a box of volume $V$, and ask for the entropy
$S(E,V)$. We let $E$ exceed the value $E_{bh}$ corresponding to the largest
black hole that can fit in the box. Several approaches in the past have
suggested the expression $S\sim \sqrt{EV/G}$. We recall these arguments, and in
particular expand on an argument that uses dualities of string theory. We
require that expression for $S(E,V)$ be invariant under the T and S dualities,
and that it agree with the black hole entropy when $E\sim E_{bh}$. These
criteria lead to the above expression for $S$. We note that this expression had
been obtained also by a imposing a quite different requirement -- that the
entropy within a cosmological horizon be of order the Bekenstein entropy for a
black hole of size the cosmological horizon. We recall the earlier proposed
model of a `dense gas of black holes' to model this entropy, and discuss its
realization as a set of intersecting brane states. Finally we speculate that
the cosmological evolution of such a phase may depart from the evolution
expected from the classical Einstein equations, since the very large value of
the entropy can lead to novel effects similar to the fuzzball dynamics found in
black holes.
