  We study the topology of the link $M^{\mathrm{trop}}_{g,n}[1]$ of the
tropical moduli spaces of curves when g=2. Tropical moduli spaces can be
identified with boundary complexes for $\mathcal{M}_{g,n}$, as shown by
Abramovich-Caporaso-Payne, so their reduced rational homology encodes
top-weight rational cohomology of the complex moduli spaces
$\mathcal{M}_{g,n}$. We prove that $M^{\mathrm{trop}}_{2,n}[1]$ is an
$n$-connected topological space whose reduced integral homology is supported in
the top two degrees only. We compute the reduced Euler characteristic of
$M^{\mathrm{trop}}_{g,n}[1]$ for all n, and we compute the rational homology of
$M^{\mathrm{trop}}_{2,n}[1]$ when $n \le 8$, determining completely the
top-weight $\mathbb{Q}$-cohomology of $\mathcal{M}_{2,n}$ in that range.
