Example of contractible open manifold that is not $\mathbb{R}^n$ - varieties?

Here is what I know about the question which leads to the conclusion that while examples mentioned by the speaker do exist, they are by no means easy; I do not know any explicit examples (it does not mean that there aren't any).

Definition. A topological manifold $M$ is called tame if $M$ is homeomorphic to the interior of a compact manifold $N$ with boundary.

The Whitehead manifold (mentioned in Thomas Rot's answer) is the first (and the most famous) example of a non-tame contractible manifold. The relevance of this definition to real algebraic sets (sets given by polynomial equations in $R^k$ for some $k$) is that every real algebraic manifold is tame. Already this is not easy, it follows from the Lojasiewicz's theorem that every compact real algebraic set admits a triangulation.

Definition. A tame manifold $M$ is simply connected at infinity if it admits a compactification $M\subset N$ as above such that $\partial N$ is simply connected.

This is not the standard definition (which is a bit tricky, I can give one if you like), but one can show it to be equivalent to the standard one; in particular, simple connectivity at infinity (which is defined not only for tame manifolds) is independent of the compactification. The following theorem is worth a triple of Fields medals (Smale, Freedman and Perelman):

Theorem. (J. Stallings, M. Freedman, G. Perelman) A contractible $n$-dimensional manifold is homeomorphic to $R^n$ if and only if it is simply connected at infinity.

Thus, in order to construct examples mentioned by the speakers, one is looking for smooth compact contractible manifolds with boundary $N$ such that $\partial N$ is not simply connected. Such manifolds $N$ do not exist in dimensions $\le 3$ (I can explain why if you like), but exist in all dimensions $n\ge 4$. (Every smooth homology $n-1$-sphere bounds a smooth compact contractible manifold, as long as $n\ge 5$; in dimension $n=4$ the examples are called Mazur manifolds, they were first constructed independently by Mazur and Poenaru around 1960.) But you need more than that: you want examples where $int(N)$ is algebraic. Akbulut and King in:

S. Akbulut and H. King, The topology of real algebraic sets with isolated singularities, Annals of Math. 113 (1981) 425-446.

proved that the interior of any smooth compact manifold (with boundary) is diffeomorphic to a nonsingular real algebraic subset of some $R^k$.

By putting it all together we obtain that for every $n\ge 4$ there exist contractible nonsingular real algebraic sets which are smooth $n$-dimensional manifolds not homeomorphic to $R^n$. On the other hand, such examples do not exist for $n\le 3$.

You are probably looking for the Whitehead manifold. I don't know of an explicit polynomial whose zero locus is the Whitehead manifold, but I'm (edit NOT)-sure that it exists.