What interests me most about the possible existence of physical infinites is their ability to predict an infinite number of universes (under the multiverse theory)
If space is indeed continuous, there could be infinite possible positions in which something could be (in exactly the same way as there are infinite numbers between 0 and 1) and for a probabilistic event, such as the position of a quantum particle, an infinite number of universes would be required (again, assuming its truth) to describe the position in spacetime of that particle alone.