  The covariant entropy bound states that the entropy, S, of matter on a
light-sheet cannot exceed a quarter of its initial area, A, in Planck units.
The gravitational entropy of black holes saturates this inequality. The entropy
of matter systems, however, falls short of saturating the bound in known
examples. This puzzling gap has led to speculation that a much stronger bound,
S< A^{3/4}, may hold true. In this note, we exhibit light-sheets whose entropy
exceeds A^{3/4} by arbitrarily large factors. In open FRW universes, such
light-sheets contain the entropy visible in the sky; in the limit of early
curvature domination, the covariant bound can be saturated but not violated. As
a corollary, we find that the maximum observable matter and radiation entropy
in universes with positive (negative) cosmological constant is of order
Lambda^{-1} (Lambda^{-2}), and not |Lambda|^{-3/4} as had hitherto been
believed. Our results strengthen the evidence for the covariant entropy bound,
while showing that the stronger bound S< A^{3/4} is not universally valid. We
conjecture that the stronger bound does hold for static, weakly gravitating
systems.
