Strongly generating elements in finite and profinite groups

Given a finite group $G$ and an element $g\in G$, we may compare the expected number $e(G)$ of elements needed to generate $G$ and the expected number $e(G,g)$ of elements of $G$ needed to generate $G$ together with $g\mathrm{.}$ We address the following question: how large can the difference $e(G)-e(G,g)$ be?