I had great luck with Acorn squash one year. Lettuce and spinach too. The squash got about 1-2 hours of direct light, and the rest was shaded by pine trees and a medium red-bud tree(actually it was planted directly under the red bud about 6' from the trunk. I got about 4-6 squashes off each of the plants throughout the summer.
Things that did NOT work so well were tomatoes, broccoli, peas, carrots, corn, potatoes. Here were my "shade crops" that I joke about when I talk about learning how to garden(don't plant in almost full shade). It is still fun to grow things anywhere though. When I started at my new house 4 years ago, there was no sun, just all overgrown property.
Shade corn pics (planted under Large tulip tree and redbuds and honeysuckles);
Shade potatoes(2 or 3 hills worth) planted under LARGE Tulip Tree;
Shade carrots(next to potatoes);