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.