Suppose that a countably infinite number of infinitely thin or perfectly symmetrical arrows is independently shot at a continuous target, with the distribution of impact points uniform over the target. (The independence requires that the arrows can go through each other--say, because they are made out of laser beams--or are removed between shots.) How probable is it that the exact center of the target will be hit by at least one arrow? In classical probability, the answer is zero. Intuitively, this is because the number of points on the target is a bigger infinity than the countably infinite number of arrows.
What if, instead, the number of arrows is greater than or equal to the number of points on the target? Unfortunately, the standard probabilistic model (a product space with the number of factors equal to the number of arrows) for the situation cannot answer that question: the probability of a point being hit by at least one of an uncountable infinity of arrows will be undefined. It would be interesting to see if there is any way of getting a non-arbitrary answer to the question outside of the standard model, say by putting some natural restrictions on which extensions of the model one allows.