I was considering the problem of expressing the roots of a general polynomial
$$ a_0 + a_1 x + ... a_n x^n$$
where $a_i, x \in \Bbb{C}$
Roots of course cannot be solely expressed using the field operations of $\Bbb{C}$ along side taking n-th roots, since each root corresponds to Quotienting the Galois Group of a given polynomial, by a cyclic group of degree $n$ and not all groups necessarily have non-trivial normal subgroups that are cyclic (namely consider $A_5$).
If we wanted however, to symbolically express the roots of polynomials, then it's clear that we would need at least one root-extraction symbol, for every finite simple group. My question then is, is this enough?
Reducing to clearer terms:
Consider a finite group $G$. Must it be the case that either $G$ is simple, or there exists a nontrivial finite simple group $L$ that is a normal subgroup of $G$?
If the answer to this is no, then its clear more radical symbols are necessary. Which leads to the next natural question:
Consider a finite group $G$, what is the minimal subset $S$ of all finite groups such that either $G$ is simple or $\exists L \in S$ such that $L$ is a non trivial normal subgroup of $G$.
A finite group need not contain a simple normal subgroup. For instance, $A_4$ does not (its only nontrivial proper normal subgroup is the subgroup of elements of order $2$, which form a Klein 4-group). On your follow-up question, it is not clear to me that there is a unique such minimal set $S$ (though there is a canonical minimal such $S$, defined inductively by saying $H$ is in $S$ iff no nontrivial proper normal subgroup of $H$ is in $S$). However, one nice such set (which may or may not be minimal) is the set of all groups which are direct products of isomorphic simple groups. In fact, every minimal (nontrivial) normal subgroup of any finite group is of this form, since a minimal normal subgroup must be characteristically simple (it has no nontrivial proper characteristic subgroups), and every finite characteristically simple group is a product of isomorphic simple groups (see here for a proof).
As a final note, however, to express roots of a polynomial in terms of intermediate operations, you are actually interested in quotients of the Galois group, not subgroups. Since every finite group obviously has a simple quotient, you should only need to worry about simple groups. The resulting expression for the roots you get by iteratively solving polynomials with simple Galois group is then just the field-theoretic mirror of a composition series of the Galois group of the original polynomial.