Finding a curve that intersects any line on the plane 
Question Is there a curve on plane  such that any line on the plane meets it (a non zero ) finite times ?

What are the bounds on the number of  such intersections.
My question was itself inspired by this "Can you draw circles on the plane so that every line intersects at least one of them but no more than 100 of them?" 
 A: Cubic parabola $$y=x^3$$ has this property. The max number of such intersections is given by the Fundamental theorem of algebra:
$$x^3=ax+b$$ can have at most 3 solutions.
A: As already shown, any cubic polynomial (and indeed, any odd-degree polynomial) has the requisite property by the fundamental theorem of algebra.
What's more, a simple perturbation argument should be enough to show that any (sufficiently) smooth curve that meets every line in at least one point will meet some lines in at least three points.  Consider a point tangent to the curve where the second derivative 'with respect to the tangent line' is non-zero; that is, a non-reflex tangent point, or locally extremal point.  (Such points must exist if the curve is non-trivial).  Now, consider pencils of lines 'near' this intersection point; displaced infinitesimally one way from the tangent, they must have another point of intersection with the curve, and this point can be made 'generic' so that it doesn't vanish under small perturbations.  Then displace infinitesimally the other direction; the 'generic' point of intersection is still a point of intersection, but the tangent turns into two points of intersection.
