  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.
