I recently stumbled over the example in http://ysharifi.wordpress.com/2010/03/09/a-uniquely-divisible-non-abelian-group/ of a non-abelian group $G$ with the property that for all natural numbers $n$ and elements $x\in G$ there is $y\in G$ such that $x=y^n$. (In this particular example $y$ is unique but I don't care about that.) I will call such groups divisible, although I'm not sure if this term is used for abelian groups exclusively.
I wonder wether there are examples of non-abelian divisible groups, that satisfy additional assumptions. I am in particular interested in non-abelian divisible
$\bullet$ simple groups,
$\bullet$ finitely generated groups,
$\bullet$ finitely presented groups,
$\bullet$ groups satisfying all or some of the above properties at the same time.
Unfortunately I don't manage to find examples, but I'd appreciate any help.