I am looking for an easy-to-remember (and non-trivial) example that vividly illustrates that the "uncorrelatedness" (in the sense of Pearson) of two random variables $X, Y$ does not imply that $X$ and $Y$ are independent. By "non-trivial" I mean that all the joint probabilities are positive (whenever the associated marginals are). I realize that it may be too tall an order to come up with a non-trivial example that is sufficiently vivid or easy-to-remember, in which case, the non-triviality condition may be relaxed.
(Whether the example features a discrete or a continuous distribution is not important per se; what matters is that the example be simple enough to think through, ideally in one's head.)
Thanks!