Is there anything special about the classes of affine varieties in the Grothendieck ring of varieties (over $\mathbb{C})$?. Is there some specialisation that allows us to distinguish classes of affine varieties from general classes?

Is there anything special about the classes of affine varieties in the Grothendieck ring of varieties (over $\mathbb{C})$?. Is there some specialisation that allows us to distinguish classes of affine varieties from general classes?

After R. van Dobben de Bruyn's answer, it might be more interesting to consider the modified version: is there anything special about classes that can be represented by an irreducible smooth affine variety?