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If I understand it correctly, this paper of Abhyankhar says that what you want is not achievable.

Here is what Abhyankar shows. I warn you that I haven't read the paper, and am just skimming the introduction. Let $K$ be the algebraic closure of $k(X,Y)$. So an element $z$ of $K$ should be thought of as a root of a polynomial $z^n + a_{n-1}(X,Y) z^n + \cdots + a_{0}(X,Y)$ whose coefficients depend on two parameters $x$ and $y$.

Let $L^1$ be the subfield of $K$ generated by all solutions to polynomials of the form $z^n + a_{n-1}(T) z^{n-1} + \cdots + a_{0}(T)$ where $T \in k(X,Y)$ and the $a_i$ are polynomials. This is pretty close to the class of $z$'s you refer to in your update, except that you ask for $T$ and the $a_i$ to be smooth and Abhyankar takes them to be rational functions. Let $L^2$ be the subfield of $K$ generated by solutions to polynomials of the form $z^n + a_{n-1}(T) z^{n-1} + \cdots + a_{0}(T)$ where $T \in L^1$ and the $a_i$ are polynomials. Inductively, make $L^3$, $L^4$, etcetera. Abhyankar shows that $\bigcup L^i$ is strictly smaller than $K$.

In fact, the meat of Abyhankar's argument is about formal power series, not polynomials. Here I had a little trouble following Abyhankar, so I hope I am summarizing him correctly. Let $\hat{A}$ be the ring of formal power series $k[[X,Y]]$ and let $\hat{K} =\mathrm{Frac} \ \hat{A}$. Let $\hat{L}^1$ be the field generated by all roots of polynomials $\sum a_i(T) z^i$ where $a_i$ are formal power series and $T$ is in the maximal ideal of $\hat{A}$. Let $\hat{B}^1$ be the integral closure of $\hat{A}$ in $\hat{L}^1$. Let $\hat{L}^2$ be the field generated by all roots of polynomials $\sum a_i(T) z^i$ where $a_i$ are formal power series and $T$ is in the maximal ideal of $\hat{B}^1$. Let $\hat{B}^2$ be the integral closure of $\hat{A}^1$ in $\hat{L}^2$. In this manner, define $\hat{L}^j$ for all $j$. Then $\bigcup \hat{L}^j$ is smaller than the algebraic closure of $\hat{K}$.

The essence of the proof is that (1) $\hat{L}^{j+1}$ is abeliansolvable1 over $\hat{L}^j$ and (2) the splitting field of $z^6+Xz+Y$ is not solvable over $\mathrm{Frac} \ k[[X,Y]]$. If you've never thought about these issues before, part (1) may surprise you. To get some intuition for this, note that the root of $x^2 - (1+t)$ is degree $2$ over $\mathrm{Frac} \ k[t]$ but this polynomial factors as $\left( x - \sum \binom{1/2}{j} t^j \right)\left( x + \sum \binom{1/2}{j} t^j \right)$ over $\mathrm{Frac} \ k[[t]]$. So passing to power series can make Galois groups smaller.

You might enjoy thinking about the topology in part (2). Here is as far as I have gotten. The polynomial $z^6+Xz+Y$ has a multiple root if and only if $5^5 X^6 + 6^6 Y^5=0$. This is a real $2$-fold in $\mathbb{C}^2$, with a singularity at $(0,0)$. Its intersection with a sphere around the origin is a torus knot of type $(6,5)$. The roots of $z^6+Yz+X$ form a $6$-sheeted cover of this torus-knot complement. Abyhankar's claim is that the monodromy of this cover is not solvable.

So, Abhyankar is disproving your hope not just for polynomials but for formal power series. The map "take the Taylor series" is a map from the ring of smooth functions to the ring of formal power series. (This map has a kernel, containing things like $e^{-1/x^2}$.) I think (but have not checked in detail) that this map should mean that Abhyankar's result also works for smooth functions.

1 I think that, at $j=0$, this statement is only right if we take $k$ algebraically closed, although I don't see where Abhyankar says this. But I'm perfectly willing to do that. Note that $\hat{L}^1$ contains the algebraic closure of $k$, so the issue of whether or not $k$ is algebraically closed goes away once we get one step up the tower. Also, I think that when $k=\mathbb{C}$, we should be able to strengthen solvable to abelian.

1

If I understand it correctly, this paper of Abhyankhar says that what you want is not achievable.

Here is what Abhyankar shows. I warn you that I haven't read the paper, and am just skimming the introduction. Let $K$ be the algebraic closure of $k(X,Y)$. So an element $z$ of $K$ should be thought of as a root of a polynomial $z^n + a_{n-1}(X,Y) z^n + \cdots + a_{0}(X,Y)$ whose coefficients depend on two parameters $x$ and $y$.

Let $L^1$ be the subfield of $K$ generated by all solutions to polynomials of the form $z^n + a_{n-1}(T) z^{n-1} + \cdots + a_{0}(T)$ where $T \in k(X,Y)$ and the $a_i$ are polynomials. This is pretty close to the class of $z$'s you refer to in your update, except that you ask for $T$ and the $a_i$ to be smooth and Abhyankar takes them to be rational functions. Let $L^2$ be the subfield of $K$ generated by solutions to polynomials of the form $z^n + a_{n-1}(T) z^{n-1} + \cdots + a_{0}(T)$ where $T \in L^1$ and the $a_i$ are polynomials. Inductively, make $L^3$, $L^4$, etcetera. Abhyankar shows that $\bigcup L^i$ is strictly smaller than $K$.

In fact, the meat of Abyhankar's argument is about formal power series, not polynomials. Here I had a little trouble following Abyhankar, so I hope I am summarizing him correctly. Let $\hat{A}$ be the ring of formal power series $k[[X,Y]]$ and let $\hat{K} =\mathrm{Frac} \ \hat{A}$. Let $\hat{L}^1$ be the field generated by all roots of polynomials $\sum a_i(T) z^i$ where $a_i$ are formal power series and $T$ is in the maximal ideal of $\hat{A}$. Let $\hat{B}^1$ be the integral closure of $\hat{A}$ in $\hat{L}^1$. Let $\hat{L}^2$ be the field generated by all roots of polynomials $\sum a_i(T) z^i$ where $a_i$ are formal power series and $T$ is in the maximal ideal of $\hat{B}^1$. Let $\hat{B}^2$ be the integral closure of $\hat{A}^1$ in $\hat{L}^2$. In this manner, define $\hat{L}^j$ for all $j$. Then $\bigcup \hat{L}^j$ is smaller than the algebraic closure of $\hat{K}$.

The essence of the proof is that (1) $\hat{L}^{j+1}$ is abelian1 over $\hat{L}^j$ and (2) the splitting field of $z^6+Xz+Y$ is not solvable over $\mathrm{Frac} \ k[[X,Y]]$. If you've never thought about these issues before, part (1) may surprise you. To get some intuition for this, note that the root of $x^2 - (1+t)$ is degree $2$ over $\mathrm{Frac} \ k[t]$ but this polynomial factors as $\left( x - \sum \binom{1/2}{j} t^j \right)\left( x + \sum \binom{1/2}{j} t^j \right)$ over $\mathrm{Frac} \ k[[t]]$. So passing to power series can make Galois groups smaller.

So, Abhyankar is disproving your hope not just for polynomials but for formal power series. The map "take the Taylor series" is a map from the ring of smooth functions to the ring of formal power series. (This map has a kernel, containing things like $e^{-1/x^2}$.) I think (but have not checked in detail) that this map should mean that Abhyankar's result also works for smooth functions.

1 I think that, at $j=0$, this statement is only right if we take $k$ algebraically closed, although I don't see where Abhyankar says this. But I'm perfectly willing to do that. Note that $\hat{L}^1$ contains the algebraic closure of $k$, so the issue of whether or not $k$ is algebraically closed goes away once we get one step up the tower.