The question was edited several times. Most recent version, suggested by Fedja:

Does there exist an open set $U\subset \mathbb R^n$

`(n>1)`

that contains balls of arbitrarily large radius and such that no polynomial mapping $p\colon \mathbb R^n\to\mathbb R^n$ takes $U$ onto $\mathbb R^n$? (Take $n=2$ if you prefer.)

Older version (with bounty) was answered by Fedja in the affirmative:

Does there exist a topological ball $U\subset \mathbb R^n$ of infinite volume such that no polynomial mapping $p\colon \mathbb R^n\to\mathbb R^n$ takes $U$ onto $\mathbb R^n$?

Discussion:

Motivated by a recent question, I wonder if there is a geometric characterization of open sets $U\subset \mathbb R^n$ that can be mapped onto $\mathbb R^n$ by a polynomial $p$. Let $U$ be a topological ball to simplify matters. The following *tail volume condition* is necessary for the existence of such $p$.

(TVC) $\int_1^{\infty} r^{m} |U\setminus B(r)|=\infty$ for some $m>0$.

Indeed, the absolute value of the Jacobian of $p$ must have infinite integral over $U$. Since the Jacobian is a polynomial, $\int_U |x|^N dx=\infty$ for some $N$. The latter can be rephrased in terms of tail volume: $\int_1^{\infty} r^{N-1} |U\setminus B(r)| =\infty$. The example of $U=\{(x,y)\in \mathbb R^2\colon x>1, 0<y<x^{-M}\}$ shows that (TVC) is somewhat sharp. This particular $U$ can be mapped onto $\mathbb R^2$ by $(x,y)\mapsto (x,x^{M+1}y)$ followed by translation and the power map $(x+iy)\mapsto (x+iy)^8$.

I am interested in other obstructions besides small tail volume, as well as in reasonably general sufficient conditions. Initially I hoped to get a stronger necessary condition from an affirmative answer to the question below, but Bjorn Poonen answered it in the negative.

"If $f\colon U\to\mathbb R^n$ is a polynomial surjection, does there exist $\epsilon>0$ such that $p(U\cap B(r))$ contains $B(r^\epsilon)$ for large $r$?"