  While there exist theories that have states "more strongly entangled" than
quantum theory, in the sense that they show CHSH values above Tsirelson's
bound, all known examples of such theories have a strictly smaller set of
measurements. Therefore, in tasks which require both bipartite states and
measurements, they do not perform better than QM. One of the simplest
information processing tasks involving both bipartite states and measurements
is that of entanglement swapping. In this paper, we study entanglement swapping
in generalised probabilistic theories (GPTs). In particular, we introduce the
iterated CHSH game, which measures the power of a GPT to preserve non-classical
correlations, in terms of the largest CHSH value obtainable after $n$ rounds of
entanglement swapping. Our main result is the construction of a GPT that
achieves a CHSH value of $4$ after an arbitrary number of rounds. This
addresses a question about the optimality of quantum theory for such games
recently raised in the literature. One challenge faced when treating this
problem is that there seems to be no general framework for constructing GPTs in
which entanglement swapping is a well-defined operation. Therefore, we
introduce an algorithmic construction that turns a bipartite GPT into a
multipartite GPT that supports entanglement swapping, if consistently possible.
