Given a realization of a stochastic process, $x_{t_1}, x_{t_2}, \ldots, x_{t_n}$, is there a simple statistic that captures the degree to which the stochastic process is mean reverting?
For example, such a statistic would give a high value for a realization of an Ornstein-Uhlenbeck process, and a low value for Brownian motion or geometric Brownian motion.
In the quantitative finance literature, you would attempt to fit an OU process to the data to get an estimate of the mean reversion parameter ($\eta$).
$$X\sim OU(\eta,\bar{x},\sigma)\implies dX_t = \eta(\bar{X}-X_t)dt +\sigma dW_t$$
Note that Brownian Motion is simply the OU process with $\eta=0$ (in line with your expectations).
Actual estimation of $\eta$ is not trivial, but here is a paper that outlies some ways to approach measuring mean reversion in stochastic models: http://www.investmentscience.com/Content/howtoArticles/MLE_for_OR_mean_reverting.pdf