This paper is motivated by three issues associated with the supersymmetric extension of the standard model: the $��$ 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.

The discussion of the following two topics are included, (1). How large can the lightest-Higgs mass be in the supersymmetric standard models? (2). The triviality bound of the Higgs mass, with the conclusion that the SUSY 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