What groups $G$ contains an element $g$ such that $G/(g\text{ is made central})=1$ (or equivalently $[g,G]=G$)?

A necessary condition is that $G$ is a perfect group. A sufficient condition is that $G$ is a nonabelian simple group. However, it seems that neither condition is complete.

I'm happy to assume $G$ is finitely generated and finitely presented (actually, I'm happy to assume $G$ is a $3$-manifold group). Other interesting necessary and sufficient conditions are also welcome.

What groups $G$ contains an element $g$ such that $G/(g\text{ is made central})=1$ (or equivalently $[g,G]=G$, where $[g,G]:=\langle[g,h]\colon h\in G\rangle$)?

A necessary condition is that $G$ is a perfect group. A sufficient condition is that $G$ is a nonabelian simple group. However, it seems that neither condition is complete.

I'm happy to assume $G$ is finitely generated and finitely presented (actually, I'm happy to assume $G$ is a $3$-manifold group). Other interesting necessary and sufficient conditions are also welcome.

What group $G$ contains an element $g$ such that $G/(g\text{ is central})=1$ (or equivalently $[g,G]=G$)?

A necessary condition is that $G$ is a perfect group. A sufficient condition is that $G$ is a nonabelian simple group. However, it seems that neither condition is complete.

I'm happy to assume $G$ is finitely generated and finitely presented (actually, I'm happy to assume $G$ is a $3$-manifold group). Other interesting necessary and sufficient conditions are also welcome.

What groups $G$ contains an element $g$ such that $G/(g\text{ is made central})=1$ (or equivalently $[g,G]=G$)?

A necessary condition is that $G$ is a perfect group. A sufficient condition is that $G$ is a nonabelian simple group. However, it seems that neither condition is complete.

I'm happy to assume $G$ is finitely generated and finitely presented (actually, I'm happy to assume $G$ is a $3$-manifold group). Other interesting necessary and sufficient conditions are also welcome.