Does there exist a characteristically simple group, which is not a direct product of simple groups? Does there exist a characteristically simple group, which is not a direct product of simple groups? A characteristically simple group is a group without non-trivial proper characteristic subgroups. 
The only thing I know, that if such group $G$ exists, it should not be Artinian:
Suppose it is. If it has no non-trivial proper normal subgroups, then it is simple. If we have one, we can choose the minimal one (let’s denote it as $N$), according to the Zorn lemma (which can be applied, as our group is Artinian). It is not hard to see that $N$ will be simple. Then, $\langle \Pi_{\phi \in Aut(G)} \phi(N) \rangle = G$ as it is characteristic.  And as all $\phi(N)$ are normal in it and either are the same subgroup or intersect trivially (because of their minimality), that results in $G$ being the direct product of $[Aut(G):Stab_{Aut(G)}(N)]$ isomorphic copies of $N$.
However, not all groups are Artinian. And I do not know how to deal with non-Artinian case here. 

Actually, as it was pointed out in the comments $(\mathbb{Q}, +)$ is an example of such group.
It is non-simple $(\mathbb{Q}, +)$ as $(\mathbb{Z}, +) \triangleleft (\mathbb{Q}, +)$.
It is characteristically simple, as for any field $\mathbb{F}$, $Aut(\mathbb{F}, +)$ acts transitively on $\mathbb{F}\setminus\{0\}$.
It is not a direct product of any two non-trivial groups, as was proved in four different ways here: Is $(\mathbb{Q},+)$ the direct product of two non-trivial subgroups?
