Let $k = \bar{k}$ a fixed field. I would like to know if there exist hypersurfaces $X \subset \mathbb{A}_k^n$ that contain no lines. By line I really mean line, and not just rational curve.

I haven't put any restrictions on $X$, but it's still not clear to me that such things exist. Most likely they form a nonempty open subset of some Hilbert scheme if the degree is large enough.