How tall would a tower at the South Pole have to be for the top to get year-around sunshine? How tall would a tower at the South Pole have to be for the top to get year-around sunshine? No doubt the height would vary each year since the Tropic of Cancer changes latitude slightly. Perhaps assume 23.5° N for calculation's sake.

assuming spherical earth with radius $R$ and axis tilt $\mu$, from geometry reasoning we get: $$ \frac{R}{R+h}=\cos \mu \Rightarrow R\left( \frac{1}{\cos \mu} - 1 \right)=h $$ !earth

so your tower height should be $h=((1/\cos(23.5)-1)\cdot 6371 \text{ km} = 576.2 \text{ km}$; greetings to ISS

for ellipsoid-Earth the height is bit different, but the above is a fine estimate