  We present a general framework for calculating post-Minskowskian, classical,
conservative Hamiltonians for $N$ non-spinning bodies in general relativity
from relativistic scattering amplitudes. Novel features for $N>2$ are described
including the subtraction of tree-like iteration contributions and the
calculation of non-trivial many-body Fourier transform integrals needed to
construct position space potentials. A new approach to calculating these
integrals as an expansion in the hierarchical limit is described based on the
method of regions. As an explicit example, we present the
$\mathcal{O}\left(G^2\right)$ 3-body momentum space potential in general
relativity as well as for charged bodies in Einstein-Maxwell. The result is
shown to be in perfect agreement with previous post-Newtonian calculations in
general relativity up to $\mathcal{O}\left(G^2 v^4\right)$. Furthermore, in
appropriate limits the result is shown to agree perfectly with relativistic
probe scattering in multi-center extremal black hole backgrounds and with the
scattering of slowly-moving extremal black holes in the moduli space
approximation.
