  We evaluate for the first time the dark matter bispectrum of large-scale
structure at two loops in the Standard Perturbation Theory and at three loops
in the Renormalised Perturbation Theory (MPTbreeze formalism), removing in each
case the leading divergences in the integrals in order to make them
infrared-safe. We show that the Standard Perturbation Theory at two loops can
be employed to model the matter bispectrum further into the quasi-nonlinear
regime compared to one loop, up to $k_{\text{max}} \sim 0.1 \, h/{\text{Mpc}}$
at $z = 0$, but without reaching a high level of accuracy. In the case of the
MPTbreeze method, we show that its bispectra decay at smaller and smaller
scales with increasing loop order, but with smaller improvements. At three
loops, this model predicts the bispectrum accurately up to scales
$k_{\text{max}} \sim 0.17 \, h/{\text{Mpc}}$ at $z = 0$ and $k_{\text{max}}
\sim 0.24 \, h/{\text{Mpc}}$ at $z = 1$.
