Does there exist a piecewise smooth curve in $\mathbb{R}^3$ such that every plane intersects the curve at finitely many points and the number of intersection points can be arbitrary large?
If the number of intersection points for each plane is bounded then there is an example: $\gamma(t)=(t,t^3,t^5)$ intersects every plane at most five points.
Try $(x,y,z) = (t + \sin(t^2), t^2, t^3)$. For any $a,b,c$ (not all $0$) and $d$, $|a (t+\sin(t^2)) + b t^2 + c t^3 - d| \to \infty$ as $t \to \pm \infty$ so the intersections are in a finite interval. And since $a (t+\sin(t^2)) + b t^2 + c t^3 - d$ is analytic and not constant, it has finitely many zeros in a compact set. So any plane has only finitely many intersections with the curve.
The curve intersects the plane $x = \sqrt{2 m \pi}$ (where $m$ is a positive integer) when $t + \sin(t^2) = \sqrt{2 m \pi}$. For $t = \sqrt{2m \pi}+s$ that says $$s + \sin((\sqrt{2m\pi}+s)^2) = s + \sin(2 \sqrt{2m\pi} s + s^2) = 0$$ In the interval $-1/2 < s < 1/2$, $2 \sqrt{2m\pi}s +s^2$ runs from $-\sqrt{2m\pi} + 1/4$ to $+\sqrt{2m\pi} + 1/4$, and thus passes through approximately $\sqrt{2m/\pi}$ odd multiples of $\pi/2$, at which the sine is alternately $\pm 1$, and thus $ s + \sin(2 \sqrt{2m\pi} s + s^2)$ has approximately $\sqrt{2m/\pi}$ sign changes. Thus the number of intersection points is unbounded.