  The simplest extension of the Standard Model is to add a gauge singlet
scalar, $S$: the singlet extended Standard Model. In the absence of a $Z_2$
symmetry $S\rightarrow -S$ and if the new scalar is sufficiently heavy, this
model can lead to resonant double Higgs production, significantly increasing
the production rate over the Standard Model prediction. While searches for this
signal are being performed, it is important to have benchmark points and models
with which to compare the experimental results. In this paper we determine
these benchmarks by maximizing the double Higgs production rate at the LHC in
the singlet extended Standard Model. We find that, within current constraints,
the branching ratio of the new scalar into two Standard Model-like Higgs bosons
can be upwards of $0.76$, and the double Higgs rate can be increased upwards of
30 times the Standard Model prediction.
