  There are two known classes of gravitational instantons with quadratic volume
growth at infinity, known as type ALG and ALG$^*$. Gravitational instantons of
type ALG were previously classified by Chen-Chen. In this paper, we prove a
classification theorem for ALG$^*$ gravitational instantons. We determine the
topology and prove existence of "uniform" coordinates at infinity for both ALG
and ALG$^*$ gravitational instantons. We also prove a result regarding the
relationship between ALG gravitational instantons of order $\mathfrak{n}$ and
those of order $2$.
