In how few points can a continuous curve meet all lines?

Question

Inspired by this (currently closed) question, I'm wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for $t\in\mathbb{R}$ and $x(t), y(t)$ continuous functions) which meets every line in the plane, what is the minimum number of times it must meet some line? If the curve is (twice?) differentiable then the answer is 'three' by perturbation arguments (consider lines on either side of a tangent point), but those arguments (or at least the ones that I've been able to put together) make more-or-less explicit use of the curve's smoothness. I suspect that some variant of the Jordan Curve Theorem might be used to show that the answer isn't even, but without the differentiability hypothesis I'm not even sure that the answer can't just be 'one' (EDIT: of course it can't be a maximum of one intersection per line, as Karolis Juodelė points out below: just choose a line that goes between some two points on the curve! This also obviates the question I had about point sets, so I've removed that outright). Can the perturbation argument be adapted to show that the minimum is three even for non-smooth curves, or is there some 'exotic' curve that meets each line no more than twice?