An example of a residually finite group which is not Hopf trying to think of any residually finite group which is not Hopf.  Any help?

As Dietrich Burde has pointed out, you need to consider infinitely generated groups. This is because of the following result due to Mal'cev.
Theorem (Mal'cev):
A finitely generated, residually finite group is Hopfian.
Proof:
Suppose $G$ is residually finite and non-Hopfian, and we shall find a contradiction. As $G$ is non-Hopfian, there exists a surjective homomorphism $\phi: G\rightarrow G$ such that $\ker\phi\neq 1$. Consider $g\in\ker\phi\setminus\{1\}$. Then as $G$ is residually finite there exists a subgroup $N_g\unlhd_fG$ such that $g\not\in N_g$, with associated map $\psi: G\rightarrow K=G/N_g$. As $G$ is finitely generated there are only finitely many, $n$ say, homomorphisms $G\rightarrow K=G/N_g$. We shall denote these homomorphisms by $\psi_1, \ldots, \psi_n$. Since $\phi$ is surjective, all of the maps $\phi\psi_i$ are distinct and so they constitute all the homomorphisms from $G$ to $K$. However, $\phi\psi_i(g)=_K1$ for all $i$ but $\psi(g)\neq 1$. This is a contradiction, as required.
Note that residual finiteness is a very strong property for finitely presented groups, but relaxing finite presentability to finite generation looses you some properties, and relaxing finite generation loses you even more. For example, a finitely presented, residually finite group has soluble word problem but there exists finitely generated, residually finite groups which are not even recursively presentable! Finite generation gives you Hopfian and also a residually finite automorphism group, but again relaxing this looses these conditions.
