The following is a generalization of a graph theory problem I have seen some work on. I would be interested to know if this question has a name, or if there has been any work outside of the graph-theoretic context.
First, let $G$ be a finitely-generated group, with $S$ a finite generating set. There should probably be the condition that $S$ is somehow "linearly independent."
Now, given a $g\in G$, we denote $r(g,S)$ to be the minimum word length over all expressions of $g$ from elements in $S$. Then define $r(G,S) = \max_{g\in G} r(g,S)$. We can think of $r(G,S)$ as a measure of how expensive it is to use $S$ in generating $G$.
Some questions might be: what does $r(G,S)$ tell us about the relationship between $S$ and $G$? What are the possible values of $r(G,S)$ over different choices of $S$? Do there exist bounds for $r(G,S)$ independent of $S$? Are there easy ways of computing $r(G,S)$?
The context I have seen this problem in is as follows. Let $\Gamma = (V,E)$ be a simple connected graph where $V = \{v_1,\dots,v_n\}$. Then we have the setup
$$G = S_n; \quad S = \{(i~j):(v_i,v_j)\in E\}$$
or
$$G = S_n; \quad S = \{(i_1,\dots i_k):(v_{i_1},\dots, v_{i_k})\text{ is a cycle or edge in }\Gamma\}.$$
For some examples, we have that $r(S_n, S)=3$ when $S$ is of the first type and $\Gamma = C_4$, and that $r(S_n, S)=4$ when $S$ is of the first type and $\Gamma = Q_3$, the cube graph. This paper and this paper give more results when $S$ is of the first type. Moreover, this paper shows that determining whether or not $r(g,S)\le k$ for $k\ge 3$ is NP-complete.