  In this paper we study the observational constraints that can be imposed on
the coupling parameter, $\hat \alpha$, of the regularized version of the
4-dimensional Einstein-Gauss-Bonnet theory of gravity. We use the scalar-tensor
field equations of this theory to perform a thorough investigation of its
slow-motion and weak-field limit, and apply our results to observations of a
wide array of physical systems that admit such a description. We find that the
LAGEOS satellites are the most constraining, requiring $| \hat \alpha |
\lesssim 10^{10} \,{\rm m}^2$. This constraint suggests that the possibility of
large deviations from general relativity is small in all systems except the
very early universe ($t<10^{-3}\, {\rm s}$), or the immediate vicinity of
stellar-mass black holes ($M\lesssim100\, M_{\odot}$). We then consider
constraints that can be imposed on this theory from cosmology, black hole
systems, and table-top experiments. It is found that early universe inflation
prohibits all but the smallest negative values of $\hat \alpha$, while
observations of binary black hole systems are likely to offer the tightest
constraints on positive values, leading to overall bounds $0 \lesssim \hat
\alpha \lesssim 10^8 \, {\rm m}^2$.
