  Is there a quantum many-body system that scrambles information as fast as a
black hole? The Sachev-Ye-Kitaev model can saturate the conjectured bound for
chaos, but it requires random all-to-all couplings of Majorana fermions that
are hard to realize in experiments. Here we examine a quantum spin model of
randomly oriented dipoles where the spin exchange is governed by dipole-dipole
interactions. The model is inspired by recent experiments on dipolar spin
systems of magnetic atoms, dipolar molecules, and nitrogen-vacancy centers. We
map out the phase diagram of this model by computing the energy level
statistics, spectral form factor, and out-of-time-order correlation (OTOC)
functions. We find a broad regime of many-body chaos where the energy levels
obey Wigner-Dyson statistics and the OTOC shows distinctive behaviors at
different times: Its early-time dynamics is characterized by an exponential
growth, while the approach to its saturated value at late times obeys a power
law. The temperature scaling of the Lyapunov exponent $\lambda_L$ shows that
while it is well below the conjectured bound $2\pi T$ at high temperatures,
$\lambda_L$ approaches the bound at low temperatures and for large number of
spins.
