  The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface
$\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the
descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The
first two terms of it are the Torelli group $\mathcal{I}_g$ and the Johnson
kernel $\mathcal{K}_g$. By a fundamental result of Johnson (1985),
$\mathcal{K}_g$ is the subgroup of $\mathrm{Mod}_g$ generated by all Dehn
twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the
group $\mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top
homology group $H_{2g-3}(\mathcal{K}_g)$ is not finitely generated. In fact, we
show that it contains a free abelian subgroup of infinite rank, hence, the
vector space $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is infinite-dimensional.
Moreover, we prove that $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is not finitely
generated as a module over the group ring $\mathbb{Q}[\mathcal{I}_g]$.
