  The upper bound on the mass of the lightest Higgs boson is provided in the
supersymmetric standard model with multi-Higgs doublets, up to two-loop order.
Relatively large corrections are expected from the experimentally unconstrained
extra Yukawa couplings. We calculate it including the two-loop leading-log
contributions as a function of the scale $\Lambda$, below which the theory
remains perturbative. Even for $\Lambda=10^{4}$ GeV, we obtain the upper bound
of 140 GeV, which is heavier than that of the MSSM by 10 GeV.
