  We consider a Laurent series defined by infinite products $g_u(t) =
\prod_{n=0}^\infty (1 + ut^{-2^n})$, where $u\in \mathbb{F}$ is a parameter and
$\mathbb{F}$ is a field. We show that for all
$u\in\mathbb{Q}\setminus\{-1,0,1\}$ the series $g_u(t)$ does not satisfy the
$t$-adic Littlewood conjecture. On the other hand, if $\mathbb{F}$ is finite
then $g_u(t)\in \mathbb{F}((t^{-1}))$ is either a rational function or it
satisfies the $t$-adic Littlewood conjecture.
