  The 3D distribution of galaxies encodes detailed cosmological information on
the expansion and growth history of the Universe. We present the first
cosmological constraints that exploit non-Gaussian cosmological information on
non-linear scales from galaxy clustering, inaccessible with current standard
analyses. We analyze a subset of the BOSS galaxy survey using ${\rm
S{\scriptsize IM}BIG}$, a new framework for cosmological inference that
leverages high-fidelity simulations and deep generative models. We use two
clustering statistics beyond the standard power spectrum: the bispectrum and a
convolutional neural network based summary of the galaxy field. We infer
constraints on $\Lambda$CDM parameters, $\Omega_b$, $h$, $n_s$, $\Omega_m$, and
$\sigma_8$, that are 1.6, 1.5, 1.7, 1.2, and 2.3$\times$ tighter than power
spectrum analyses. With this increased precision, we derive constraints on the
Hubble constant, $H_0$, and $S_8 = \sigma_8 \sqrt{\Omega_m/0.3}$ that are
competitive with other cosmological probes, even with a sample that only spans
10% of the full BOSS volume. Our $H_0$ constraints, imposing the Big Bang
Nucleosynthesis prior on the baryon density, are consistent with the early time
constraints from the cosmic microwave background (CMB). Meanwhile, our $S_8$
constraints are consistent with weak lensing experiments and similarly lie
below CMB constraints. Lastly, we present forecasts to show that future work
extending ${\rm S{\scriptsize IM}BIG}$ to upcoming spectroscopic galaxy surveys
(DESI, PFS, Euclid) will produce leading $H_0$ and $S_8$ constraints that
bridge the gap between early and late time measurements and shed light on
current cosmic tensions.
