If I can truly only have one pair of shoes, it seems it would have to be stylish, appropriate for all events, and utilitarian.
Given this, I would choose something similar to my Polo McCallum Chukka boots in dark brown shell cordovan. They are stylish and also versatile enough that they can be worn with anything from jeans to a suit. They are comfortable and the shell cordovan makes them durable. Also, instead of a leather sole they have a Dainite sole, which adds to the comfort factor if I have to do a lot of walking and allows me to wear them on rainy days. I would look fairly silly walking around in them with shorts during the summer, but I'm trying to play within the rules here.