Example of Noetherian group (every subgroup is finitely generated) that is not finitely presented

Tarski monsters provide examples of 2-generator noetherian groups that is not finitely presented.

Edit (YCor): Tarski monsters, as defined in the link (infinite groups of prime exponent $p$ in which every nontrivial proper is cyclic) exist for large $p$ and all currently known constructions of Tarski monsters are known to yield groups that are not finitely presented. However, it is unknown whether there exists a finitely presented Tarski monster.

It's unknown whether every slender group is virtually polycyclic. See page 87 of Matt Clay's thesis.

EDIT: Primoz rightly points out that a Tarski monster is slender (and not finitely presentable!). This seems right. I'm not sure what to make of Clay's claim (which I'm fairly sure I've seen elsewhere). Presumably it's unknown whether there are finitely presented, non-virtually-polycyclic, slender groups. As James points out, one can impose other conditions, like residual finiteness, that rule out such pathological examples.