From a post to The Jouanolou trick:

Are all topologically trivial (contractible) complex algebraic varieties other then affine lines necessarily affine? Are all of them rational?

The examples that come to my mind are like a singular $\mathbb P^1$ without a point given by equation $x^2 = y^3$. This particular curve is also birationally equivalent to affine line.

Perhaps the affine part would follow from a comparison between Zariski cohomology and complex cohomology?