Although it doesn't directly answer your question, it seems interesting to consider groups in which every normal generating set is already a generating set.
At least in the finitely generated world, this is equivalent to "every maximal subgroup is normal", which is also equivalent to $G' \leq \Phi(G)$. For finite groups, this is equivalent to being nilpotent., but for finitely generated groups it may be a strictly weaker condition than nilpotence (I don't know an example however.)
In any case, such a group (for example, any f.g. nilpotent group) necessarily has $\mathrm{nr}(G) = \mathrm{rank}(G)$.
