Poking through the DGS book mentioned in Ian's answer I came along a reference to a paper that turns out to be exactly what I wanted when I asked this question originally, so I'll post it here for the sake of closure and because it's a nice example.

The paper is by Wolfgang Krieger: *On the uniqueness of the equilibrium state*, Mathematical Systems Theory **8** (2), 1974, p. 97-104.

The example is the Dyck shift, which is easiest to understand in terms of brackets. The alphabet of the shift is a collection of $2n$ symbols that come in $n$ pairs; each pair has a left element and a right element. So with $n=2$ we can write the four symbols as ( ) [ ]. The shift space $X$ comprises all sequences on these symbols in which the brackets are "opened and closed in the right order". So for example, ( ) [ ] is a legal word, as is ( ( ( ) [ ] [, but ( [ [ ) is illegal because the ( bracket cannot be closed before the [ brackets are.

Sticking with $n=2$, let $B_-\subset X$ be the set of all sequences in which every left bracket has a corresponding right bracket, and $B_+$ be the set of all sequences in which every right bracket has a corresponding left bracket. One can show that every shift-invariant measure has $\mu(B_- \cup B_+) = 1$ by partitioning the complement into a countable collection of disjoint sets indexed by the location of the first/last left/right bracket with no partner.

Define a map $\pi_+\colon B_+ \to \{0,1,2\}^\mathbb{Z}$ by sending ( to 0, [ to 1, and both ) and ] to 2. Then $\pi_+$ is an isomorphism between the two shift spaces because every right bracket has a corresponding left bracket, and hence its identity as ) or ] is uniquely determined by the rules of the shift. Similarly, the analogous map $\pi_- \colon B_- \to \{0,1,2\}^\mathbb{Z}$ is an isomorphism.

Because every ergodic invariant measure on $X$ is supported on either $B_-$ or $B_+$, we conclude that $h(X) = \log 3$ and that there are exactly two ergodic measures of maximal entropy $\mu_{\pm} = \nu \circ \pi_{\pm}$, where $\nu$ is the $(\frac 13, \frac 13, \frac 13)$-Bernoulli measure on the full 3-shift. Each of these measures gives positive measure to every open set in $X$, and each is of positive entropy -- indeed, each is Bernoulli, which is part of what makes this answer so satisfying to me.

Note that for larger values of $n$ the same argument shows that $h(X) = \log(n+1)$.