I'll toss in my two bits here.
Note that profile = tire sidewall height / tire width. A wider tire is going to have a lower profile number even if the sidewall height is the same as a narrower tire. So it would seem that a better question would be the ideal rim size for a given width and outer diameter tire.
Tire construction can compensate for sidewall height. Indy Racing League cars have tall but very stiff sidewalls, and the slip angles involved at the limits for IRL cars are less than that of Formula 1 cars which in turn are less than the lower classes of racing cars.
One issue with "low profile" tires, is the larger wheels and the greater amount of weight, increasing unsprung weight. Since the key factor is ratio of unsprung weight to total weight the weight of the car is a factor.
I can only think of two examples: Light cars like a Formula Ford, Formula Mazda (Rousch), Caterham, seem to do best with 13 inch wheels, and fairly tall sidewalls. Most Porsche 911 track cars are running 18 inch wheels, with relatively shorter sidewalls, but still smaller than the 19 inch wheels you find on some street versions.
Formula 1 cars also have smallish wheels and tall sidewalls, but I don't know the actual numbers.