  The recent results from the ATLAS and CMS collaborations show that the
allowed range for a Standard Model Higgs boson is now restricted to a very thin
region. Although those limits are presented exclusively in the framework of the
SM, the searches themselves remain sensitive to other Higgs models. We recast
the limits within a generic supersymmetric framework that goes beyond the usual
minimal extension. Such a generic model can be parameterised through a
supersymmetric effective Lagrangian with higher order operators appearing in
the K\"ahler potential and the superpotential, an approach whose first
motivation is to alleviate the fine-tuning problem in supersymmetry with the
most dramatic consequence being a substantial increase in the mass of the
lightest Higgs boson as compared to the minimal supersymmetic model. We
investigate in this paper the constraints set by the LHC on such models. We
also investigate how the present picture will change when gathering more
luminosity. Issues of how to combine and exploit data from the LHC dedicated to
searches for the standard model Higgs to such supersymmetry inspired scenarios
are discussed. We also discuss the impact of invisible decays of the Higgs in
such scenarios.
