  We study the list chromatic number of the Cartesian product of any graph $G$
and a complete bipartite graph with partite sets of size $a$ and $b$, denoted
$\chi_\ell(G \square K_{a,b})$. We have two motivations. A classic result on
the gap between list chromatic number and the chromatic number tells us
$\chi_\ell(K_{a,b}) = 1 + a$ if and only if $b \geq a^a$. Since
$\chi_\ell(K_{a,b}) \leq 1 + a$ for any $b \in \mathbb{N}$, this result tells
us the values of $b$ for which $\chi_\ell(K_{a,b})$ is as large as possible and
far from $\chi(K_{a,b})=2$. In this paper we seek to understand when
$\chi_\ell(G \square K_{a,b})$ is far from $\chi(G \square K_{a,b}) = \max
\{\chi(G), 2 \}$. It is easy to show $\chi_\ell(G \square K_{a,b}) \leq
\chi_\ell (G) + a$. In 2006, Borowiecki, Jendrol, Kr\'al, and Miskuf showed
that this bound is attainable if $b$ is sufficiently large; specifically,
$\chi_\ell(G \square K_{a,b}) = \chi_\ell (G) + a$ whenever $b \geq
(\chi_\ell(G) + a - 1)^{a|V(G)|}$. Given any graph $G$ and $a \in \mathbb{N}$,
we wish to determine the smallest $b$ such that $\chi_\ell(G \square K_{a,b}) =
\chi_\ell (G) + a$. In this paper we show that the list color function, a list
analogue of the chromatic polynomial, provides the right concept and tool for
making progress on this problem. Using the list color function, we prove a
general improvement on Borowiecki et al.'s 2006 result, and we compute the
smallest such $b$ exactly for some large families of chromatic-choosable
graphs.
