The Stone-Weierstrass Theorem guarantees that every continuous function defined on a closed interval [a,b] can be approximated as closely as desired by a polynomial.
I'm curious in polynomials of the form $P(x) = a_0 + a_1 f(x)^1 + a_2 f(x)^2 + a_3 f(x)^3$ where $f(x)$ is some continuous function f(x) could equal $e^x$ for example. What can be said for polynomials of this form? Does something similar to the Stone-Weierstrass Theorem hold?
On
You seem to be a bit confused about what the Stone-Weierstrass theorem is. The Weierstrass approximation theorem (not the Stone-Weierstrass theorem!) says that any continuous function on a closed interval is a uniform limit of polynomials. The Stone-Weierstrass theorem is a vast generalization of the Weierstrass approximation theorem, and (one version of it) says that for any compact Hausdorff space $X$, any unital subalgebra of $C(X,\mathbb{R})$ that separates points is dense. In particular, if $f$ is any injective function, the unital subalgebra it generates (i.e., the set of polynomials in $f$) is dense.
In other words, when $f$ is injective on $[a,b]$, the Stone-Weierstrass theorem itself says that every continuous function on $[a,b]$ is a uniform limit of polynomials in $f$. (Note that the assumption that $f$ is injective is obviously necessary, since if $f(c)=f(d)$ then the same will be true of any uniform limit of polynomials in $f$).
If $f$ is injective and continuous, then the Stone-Weierstrass allows us to deduce that the functions of the form $\sum_{k=0}^na_kf^k$ are dense in $C\bigl([a,b]\bigr)$. Actually, assuming that $f$ is continuous, the set that I described is dense if and only if $f$ is injective.