Naturalness bounds in extensions of the MSSM without a light Higgs boson

Adopting a bottom-up point of view, we make a comparative study of the simplest extensions of the MSSM with extra tree level contributions to the lightest Higgs boson mass. We show to what extent a relatively heavy Higgs boson, up to 200-350 GeV, can be compatible with data and naturalness. The price to pay is that the theory undergoes some change of regime at a relatively low scale. Bounds on these models come from electroweak precision tests and naturalness, which often requires the scale at which the soft terms are generated to be relatively low.