Strongly generating elements in finite and profinite groups

Given a finite group G and an element g∈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. We address the following question: how large can the difference e(G)−e(G,g) be? We prove that in general this difference can be arbitrarily large. For example for every positive integer n there exists a finite 2-generated group G such that e(G)≥n but e(G,g)≤5 for some g∈G. However, if the derived subgroup of G is nilpotent, then e(G)−e(G,g)≤11 for every g∈G.