Parissies Squares comes to mind, as do various track and field events. If the English settled outside of Earth as vigorously as they colonized it at their height, it might include cricket. Perhaps handball would feature; it seems to be popular among many species.
The great difficulty with Olympics in the Trek world is the disparity between various species....humans are more or less the same, but Vulcans would have a decided advantage over human competitors, some species might be semiaquatic, and then there's the fact that different species come from planets with different gravities, so some would be at a disadvantage and some at an advantage.