  We study the implications for the minimal supersymmetric standard model
(MSSM) of the absence of a direct discovery of a Higgs boson at LEP. First we
exhibit 15 physically different ways in which one or more Higgs bosons lighter
than the LEP limit could still exist. For each of these cases -- as well as the
case that the lightest Higgs eigenstate is at, or slightly above, the current
LEP limit -- we provide explicit sample configurations of the Higgs sector as
well as the soft supersymmetry breaking Lagrangian parameters necessary to
generate these outcomes. We argue that all of the cases seem fine-tuned, with
the least fine-tuned outcome being that with Higgs mass near 115 GeV. Seeking
to minimize this tuning we investigate ways in which the ``maximal-mixing''
scenario with large top-quark trilinear A-term can be obtained from simple
string-inspired supergravity models. We find these obvious approaches lead to
heavy gauginos and/or problematic low-energy phenomenology with minimal
improvement in fine-tuning.
