I don't know. Maybe a day or two.
When I was little I'd often fast in order to concentrate on a drawing, lego model, or something of that nature. I'm not sure why, but I always felt I was more creative and talented when hungry.
My dad has done 40 day fasts (I think) 3 times. The first time it was a complete fast, with nothing but water. The other times he had plenty of fruit juice and some meal replacement shakes.