We discuss the discovery reach of CERN LEP 2 for the Higgs sector of a general extension of the MSSM including a single gauge singlet field. This change introduces a new quartic Higgs boson self-coupling which can increase the masses of the {ital CP}-even states, and also allows mixing between singlet and nonsinglet states which can reduce the couplings of the mass eigenstates to the {ital Z}. The lightest {ital CP}-even Higgs boson is bounded by a parameter {Lambda} which takes a maximum value {Lambda}{sub max}{approx_equal} 136{endash}146 GeV for a top quark mass of 150{endash}195 GeV. We generalize the discussion of the bound to include the entire {ital CP}-even spectrum and show how experiment may exclude values of {Lambda} smaller than some {Lambda}{sub min}. {ital CP}-even Higgs boson searches at LEP 2 will be able to exclude {Lambda}{sub min}{approx_equal} 81{endash}105 GeV, depending on the machine parameters. We also present exclusion plots in the {ital m}{sub {ital A}}-tan{beta} plane, based on an analysis of {ital CP}-even, {ital CP}-odd, and charged Higgs boson production processes at LEP 2. {copyright} {ital 1996 The American Physical Society.}