Let’s call a group marginally simple if it does not have any non-trivial marginal subgroup (strict definition of marginal subgroups and brief overview of their properties can be found here: https://groupprops.subwiki.org/wiki/Marginal_subgroup). Does there exist some sort of classification of finite marginally simple groups?
$G^n$, with $G$ being a finite simple group, is always marginally simple as it has no nontrivial characteristic subgroups and all marginal subgroups are characteristic. Moreover, all groups that have no non-trivial characteristic subgroup (or even no non-trivial verbal subgroup, as shown here: Does there exist some sort of classification of finite verbally simple groups?) are of this form. However, maybe marginally simple groups are more abundant…
If $G$ is marginally simple it is either abelian or centerless, as center is a marginal subgroup.
If $G$ is abelian, then any marginal subgroup in it can be defined using a word $x^n$. That results in marginal subgroups of abelian groups being exactly the sets of all elements of order dividing $n$. And the only abelian groups that have no subgroups of such type are exactly $C_p^n$ for prime $p$, which makes them the only abelian marginally simple groups.
However, I do not know, how to deal with the case, when $G$ is centerless.
EDIT: Actually, there do exist finite marginally simple groups, which are not characteristically simple. One of the possible example series is $S_n$ for $n \geq 5$ (For what $n$ is $A_n$ a marginal subgroup of $S_n$?)