Why isn't the side of the earth facing the sun completely illuminated by sunlight from the north to the south pole every day of the year?
If the diameter of the Sun is $R_S$, the diameter of the Earth is $R_E$, and the orbital radius of the Earth is $r$, as in the drawing below (sorry about the quality),  then you can do some geometry and compute $\theta$, which is the angle at which the light from the edge of the Sun's disk as seen from Earth is tangent to the Earth's surface referenced to a line between the centres of the Sun and the Earth.



The expression for $\theta$ comes out to be

$$\theta = \sin^{-1}\left(\frac{R_S - R_E}{r}\right)$$

(You get this by extending the tangent line and the line connecting the centres, realising that this makes two similar right triangles with the angle at the tip being $\theta$, and the base of the larger one being $R_S$, then working out what the hypotenuse of the larger triangle is, which gives you $\sin\theta$.)

Since $\theta$ is small we can just say $\sin \theta = \theta$ in the usual way, and plugging in values we get $\theta \approx 0.0046\,\mathrm{rad}$.

If the angle at which the Earth's rotation axis is tilted is more than $\theta$, then there will be a period of the year during which the poles are not illuminated as the point at which the Sun's light is tangent to the Earth's surface is sunward of the pole.  Well, the angle of tilt is $23.5^\circ \approx 0.41\,\mathrm{rad}$, which is about $90\theta$.