  A uniform algebra $A$ on its Shilov boundary $X$ is {\em maximal} if $A$ is
not $C(X)$ and there is no uniform algebra properly contained between $A$ and
$C(X)$. It is {\em essentially pervasive} if $A$ is dense in $C(F)$ whenever
$F$ is a proper closed subset of the essential set of $A$. If $A$ is maximal,
then it is essentially pervasive and proper. We explore the gap between these
two concepts. We show the following: (1) If $A$ is pervasive and proper, and
has a nonconstant unimodular element, then $A$ contains an infinite descending
chain of pervasive subalgebras on $X$. (2) It is possible to imbed a copy of
the lattice of all subsets of $\N$ into the family of pervasive subalgebras of
some $C(X)$. (3) In the other direction, if $A$ is strongly logmodular, proper
and pervasive, then it is maximal. (4) This fails if the word \lq strongly' is
removed. We discuss further examples, involving Dirichlet algebras, $A(U)$
algebras, Douglas algebras, and subalgebras of $H^\infty(\mathbb{D})$. We
develop some new results that relate pervasiveness, maximality and relative
maximality to support sets of representing measures.
