Tapioca pudding made with rice or soy milk and no eggs, and a fruit topping

Fruit pie with a vegetable oil (or shortening, yuk) crust

Make sure any flour and milk substitutes you use are processed in nut-free facilities, if he's highly allergic.

I haven't done much eggless baking, but there are lots of recipes for vegan cakes online, and I know I don't miss the baked goods at my favorite vegan restaurant. There are links to some yummy sounding vegan cupcake recipes here - coconut oil is optional in all of them, which IME if he can tolerate coconut, adds some wonderful richness, but the recipes allow for other shortening if he can't have coconut.