The definitions below are taken from Solvability by radicals implies a radical formula for its roots (question by Eparoh):
Definition 1: We say that a field extension $F/K$ is a radical extension if we can form a chain of fields $$K=K_0 \leq K_1 \leq \cdots \leq K_n=F$$ where $K_{i+1}/K_i$ is a simple extension such that $K_{i+1}=K_i(a_i)$ and $a_i^{k_i} \in K_i$ for some positive integer $k_i$.
Definition 2: Let $K$ be a field and $f(x) \in K[x]$, we say that $f$ is solvable by radicals if there exists a radical extension $F/K$ such that $F$ contains a splitting field of $f$ over $K$.
This question has no answer, but it has a comment by reuns:
The radical formulas for the roots depend on constants of $K$, once the polynomial is fixed this is all we want (there are algorithms for the splitting field minimal polynomials and Galois group, if it is solvable we can unroll to find the radical formulas). What you are asking is if there are finitely many radical formulas $F_{d,l}$ of $d+1$ variables such that for every solvable polynomial $∑_{j=0}^{d} c_j x_j \in K[x]$ of degree $d$ its roots are given by $F_{d,l}(c_0,…,c_d)$ for some $l$. This is the problem of moduli space / parametrization of solvable polynomials of degree $d$.
Let $K$ be a field. Can you give an example of $f \in K[x]$ that is solvable by radicals but cannot be expressed by using only polynomial coefficients, $+, -, \cdot, \frac{...}{...}$ and proof of this fact?
Edit: the answer to this question is obvious and it is not what I actually wanted to ask. I forgot to specify $\sqrt[n]{...}$ as an operation that we can use in an expression. For this reason I created another question, Example of $f \in K[x]$ solvable by radicals but having a root inexpressible only by coefficients of $f$ and +, -, *, /, $\sqrt[n]{...}$ which asks the question I indeed wanted to ask.
Any polynomial that is solvable with radicals but with no roots in $K$ would work. The four arithmetic operations cannot take you outside of $K$.
Take $K=\Bbb Q$ and $f(x)=x^2-2$, for instance.