Only an extreme tiny parameter space for theories with five Higgs bosons (such as minimal and next to minimal supersymmetric models) rather than the one Higgs boson of the Standard Model remains open, and only in one specific subset of such models (type X - "lepton specific" models), for a very narrow range of parameters in that type of model.
Specifically, the parameter space requires that the charged Higgs bosons have masses of less than 200 GeV that are almost degenerate with the SM Higgs boson at 125.5 GeV, have a large "tan beta", a precise difference between the parameter angle beta and another parameter angle alpha equal to about pi/2 (i.e. 90 degrees), and a fairly light pseudoscalar Higgs boson.
The remaining two Higgs Doublet parameter space can probably be ruled out (or confirmed) in the second LHC run.
This is a really blow to SUSY theories and a fortiori, to String Theory, as well as the larger and more generic class of two Higgs Doublet models.
If two Higgs doublet models are ruled out, then any SUSY model must have at least three Higgs doublets (i.e. nine Higgs bosons) with nine of them at very high masses.