Other than mentioning the watch model perhaps we should also share why that particular model was selected. Let me start...
I picked AP RO as the "one watch" due to its versatility. Some have mentioned it does not hold very well to regular wear (as in the case can get scratched quite easily), but it is one of the first luxury sport watches in steel after all. So that covers the sporty side.
On the dressy side, I suppose it's ok to wear a Rolex sports model with a suit nowadays but still the RO has an elegant quality that is hard to find elsewhere. The Genta-designed case and integrated bracelet would go well whether dressed up or down.
The next contender would be the Lange 1 or GO Pano (to be honest I find the GO layout more attractive though it may be 'inspired' by the Lange).