  For $\beta > 1$, a sequence $(c_n)_{n \geq 1} \in \mathbb{Z}^{\mathbb{N}^+}$
with $0 \leq c_n < \beta$ is the \emph{beta expansion} of $x$ with respect to
$\beta$ if $x = \sum_{n = 1}^\infty c_n\beta^{-n}$. Defining $d_\beta(x)$ to be
the greedy beta expansion of $x$ with respect to $\beta$, it is known that
$d_\beta(1)$ is eventually periodic as long as $\beta$ is a Pisot number. It is
conjectured that the same is true for Salem numbers, but is only currently
known to be true for Salem numbers of degree 4. Heuristic arguments suggest
that almost all degree 6 Salem numbers admit periodic expansions but that a
positive proportion of degree 8 Salem numbers do not. In this paper, we
investigate the degree 6 case. We present computational methods for searching
for families of degree 6 numbers with eventually periodic greedy expansions by
studying the co-factors of their expansions. We also prove that the greedy
expansions of degree 6 Salem numbers can have arbitrarily large periods. In
addition, computational evidence is compiled on the set of degree 6 Salem
numbers with $\text{trace}(\beta) \leq 15$. We give examples of numbers with
$\text{trace}(\beta) \leq 15$ whose expansions have period and preperiod
lengths exceeding $10^{10}$, yet are still eventually periodic.
