  This paper is motivated by three issues associated with the supersymmetric
extension of the standard model: the $\mu$ problem, the possibility of raising
the upper bound on the lightest-Higgs mass, and the triviality problem
associated with the Higgs sector. A new scheme based on the Higgs-Higgs
condensation is proposed, and it is shown that these three issues are well
solved by this scheme. As the first realization of this new scheme, the Minimal
Supersymmetric Model of Higgs-Higgs Condensation (MSMHHC) is constructed and
studied in detail. The MSMHHC is identical with the MSSM (Minimal
Supersymmetric Standard Model) in the fundamental particle content, and their
lagrangians differ only in the Higgs sector. The Higgs sector of the MSMHHC is
based on the softly-broken supersymmetric Nambu$-$Jona-Lasinio model with the
four-field interaction of the Higgs doublets. At low energy, these two Higgs
doublets condense into two neutral Higgs gauge singlets, and the low-energy
effective lagrangian of the MSMHHC has the form of the non-minimal
supersymmetric standard model which contains two more Higgs singlets than the
MSSM. Another unique feature of the MSMHHC is that heavy top quark always
implies strongly-interacting low-energy Higgs sector, i.e., a large mass for
the lightest Higgs boson. A systematic study of the parameter space is also
made in order to reveal the qualitative features of the MSMHHC. Finally, we
comment on the question "How large can the lightest-Higgs mass be?" with the
conclusion that the supersymmetric model of Higgs-Higgs condensation will be
the most promising candidate if the lightest-Higgs mass of the MSSM is excluded
by the future experiments.
