This answer seems to make sense. My interpretation: A random variable and its square may be uncorrelated (since apparently lack of correlation is something like linear independence) but they are clearly dependent.
I guess an example would be that (standardised?) height and height$^2$ might be uncorrelated but dependent, but I don't see why anyone would want to compare height and height$^2$.
For the purpose of giving intuition to a beginner in elementary probability theory or similar purposes, what are some real-life examples of uncorrelated but dependent random variables?