From a post to The Jouanolou trick:

Are all

topologically trivial(contractible)complex algebraic varietiesnecessarily affine? Are there examples of those not birationally equivalent to an affine space?

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

Perhaps the "affine" part follows from a comparison between Zariski cohomology and complex cohomology?