  We study the phenomenology of the two Higgs doublet model with a real singlet
scalar $S$ (2HDMS). The model predicts three CP-even Higgses $h_{1,2,3}$, one
CP-odd $A^0$ and a pair of charged Higgs. We discuss the consistency of the
2HDMS with theoretical as well as with all available experimental data. In
contrast with previous studies, we focus on the scenario where $h_2$ is the
Standard Model (SM) 125 GeV Higgs, while $h_1$ is lighter than $h_2$ which may
open a window for Higgs to Higgs decays. We perform an extensive scan into the
parameter space of 2HDMS of type I and explore the effect of the
singlet-doublet admixture. We found that a large singlet-doublet admixture is
still compatible with the recent Higgs data from LHC. Moreover, we show that
$h_1$ could be quasi-fermiophobic and would decay dominantly into two photons.
We also study in details the consistency of the non-detected decay of $h_2\to
h_1 h_1$ with LHC data followed by $h_1\to \gamma \gamma$ which leads to four
photons final state at LHC: $pp\to h_2\to h_1 h_1\to 4\gamma$. Using the
results of null searches of multi-photons carried by the ATLAS collaboration,
we have found that a large area of the parameter space is still allowed. We
also demonstrate that various neutral Higgs of the 2HDMS could have several
exotic decays.
