The Minimal Supersymmetric Standard Model, or MSSM, uses two Higgs doublets which – after the "eating" of 3 out of 8 real polarizations by the $W^+,W^-,Z^0$ bosons – leave 5 physical polarizations of the Higgs. Two of them are electrically charged, $H^\pm$, and three of them are neutral. One of the three neutral ones is CP-odd, $A$, and the other two are CP-even, $h$ and $H$.
The $h$ boson is the lighter one and has to be below 135 GeV or so. The $H$ boson is heavier and may be anywhere above 135 GeV or so. It's true that models where the heavier Higgs is vastly heavier are recently favored in literature, for various reasons, but that doesn't follow from supersymmetry. Supersymmetry could work with two Higgses at 125 and 244 GeV (if I borrow another 2-sigma bump from ATLAS), among many other options.
Even without SUSY, one could construct multi-Higgs-doublet models. But if we assume that the couplings are comparable, they "share" the total cross section of a single Higgs. So you couldn't produce two peaks in the gamma-gamma channel, for example, that would be equally strong as the peak of the Standard Model Higgs. They would be smaller.
The announced "glimpse" of the Higgs today does agree with the SM Higgs. Of course, it's not measured too accurately but the existence of two light Higgses that share the cross section is already close to excluded, too. Moreover, if you look at all the channels at all the colliders, it's pretty clear that there is only one Higgs. Look at Phil Gibbs' galactic tri-collider Yes/No reconstructions:
The "Yes" peak near 125 GeV is really sharp and clean now. Another Higgs well above 200 GeV or so remains viable, especially if it is at many TeVs, and its existence would modify the cross sections and properties of the light 125 GeV Higgs boson just by a few percent. The light Higgs would remain Standard-Model-like.