Famously, the roots of fourth degree polynomials and below can always be expressed in terms of radicals but those fifth degree and above cannot necessarily be so expressed.
Allow me to rephrase this. For each degree $d$, there exists a minimal set $S$ of polynomials such that the roots of a degree $d$ polynomial can always be expressed as some combination of the (multivalued) inverses of the polynomials in $S$. For degree $2$, this set is $\{x^2\}$; for degree $3$, this set is $\{x^2,x^3\}$ and for degree $4$, this set is $\{x^2,x^3\}$ (since $\sqrt[4]{x} = \sqrt{\sqrt{x}}$).
In general, we can always construct such a set that is not necessarily minimal. For example, we can just take the set of all polynomials $f$ of degree $d$ such that $f(0)=0$. Is the minimal set always finite?
If it is finite, can we understand the set using Galois theory at all? For example, maybe there is one polynomial in the set for each simple Galois group of order $\leq d$?
For degree five, $\{\,x^2,x^3,x^5,x^5+x\,\}$ is such a set. Given $a$, a solution of $x^5+x=a$ is called a Bring radical, and if you allow the operation of taking Bring radicals, you can solve any quintic in closed form. See https://en.wikipedia.org/wiki/Bring_radical for details.
For higher degrees, there are similar finite sets of polynomials, but I don't have a reference at hand.