Harnack's curve theorem tells us that the maximum number of connected components of an algebraic curve of degree $d$ in the real projective plane is $1 + (d-1)(d-2)/2$ (and this bound is sharp).
What happens if "projective" is replaced by "affine"? What is the maximum number of connected components then?
The only relevant reference I have been able to find so far is a paper by Korchagin and Weinberg on the isotopy classification of affine quartic curves, where they calculate that a compact affine quartic curve has no more than 4 connected components and a noncompact affine quartic curve has no more than 7 connected components. I think their argument can be extended to higher degree to give an upper bound, but are these bounds sharp?