The box counting dimension of a set, $X$ in a metric space looks at boxes of dimension $\varepsilon$ and how many, $N(\varepsilon)$ are needed to covert $X$ and then says
$$dim_{box}(X) = \lim_{\varepsilon \to 0}{\frac{\log{N(\varepsilon)}}{\log{\varepsilon}}}$$
Let's use $\mathbb{R}^n$ and the Euclidean metric as our metric. Can we extend this definition to unbounded sets of $\mathbb{R}^n$ where the number of boxes needed is infinite? For example, the box counting dimension of Cantor's middle third set is $\log{2}/\log{3}$. But, what if we have an countable number of copies of this (e.g. stacked above above one another at a distance of $1$).