I recently encountered the metric mean dimension, which is a numerical metric invariant of (discrete time, compact space) dynamical systems that refines topological entropy for infinite-entropy systems. I am wondering if anything similar can be found in the literature for any metric notion of dimension (Let say that by ``metric'' means bi-Lipschitz invariant).

Put another way, I have a compact metric space $X$ that has infinite dimension for any sensible notion of dimension, and I would like to make this statement quantitative. I see two ways to do this.

The first one is to mimic the box-dimension, and consider the (extra-polynomial) growth rate of the smallest number of $\varepsilon$-balls needed to cover $X$ when $\varepsilon^{-1}$ goes to infinity. This is the simplest way to go, but I am concerned by the fact that box dimension have not as nice a behavior than Hausdorff dimension (for example countable spaces can have positive box dimension).

The second one, suggested by Greg Kuperberg, is to mimic Hausdorff dimension but replacing the family of "size functions" $(x\mapsto x^s)_s$ by another family with similar properties, like $(x\mapsto\exp(-\lambda/x)_\lambda)$.

My question is the following: do you know any example of such an invariant in the literature? Where is it used, in what purpose?