Let $G$ be a finite group and $S$ be generating set it. Now given all words with alphabet $S$, then there exists a minimum word length $N(S,G)$ such that all group elements are represented by a word of length $\le N(S,G)$. A famous example is the group of Rubik's Cube where the answer is known to be 26 (if $S$ is the set of all quarter twists) and this is known as God's number.

Where can I find more example of this kind, i.e. groups where $|S|$ and $N(S,G)$ are know (or there is at least some bound) and we have that $|S|,N(S,G)\ll |G|$?

More generally, can we expect that for most groups we can find some generating set $S$ such that $|S|,N(S,G)\ll |G|$?