Cakes are harder without eggs, but something like a banna bread loaf in a tin comes out nicely. You could do a couple and sandwich together/cover with icing, to make a square cake. Or make individual muffins, they're also fine without egg.
also in Fiona cairn's the birthday cake book there's an egg, dairy and nut free cake. she fills it with fresh raspberries but use whatever you like. again this is untested by me. Personally i'd go with Gh's triple tested recipe above.
serves 12. 140ml sunflower oil, 360 g sr flour, 60g cocoa, 2 tsp baking powder, 1 tsp salt (i'd use way less than that personally), 400g caster sugar, 2 tsp vanilla extract, 4 tsp white wine vinegar, 400g raspberries (to fill cake with)
preheat oven to 170c /gas 3 1/2 oil 2 23cm tins and line bases
sift together flour, cocoa, baking powder and salt. stir in sugar.
in another bowl measure in the oil and 400 ml water and the vanilla, then add the flour mixture, stirring in the vinegar last of all. pour batter between the 2 tins and bake for 30-35 mins or until a skewer emerges clean.
when cool put one cake on a cake stand, top with most of the raspberries, place other cake on top and scatter with rest of the raspberries. Sift over some icing sugar to serve.
I baked DS1 a boiled carrot cake from Australian Woman's weekly - he got really bad eczema from eggs back then. 2nd birthday was a vegan chocolate cake. They're a bit puddingy, but perfectly delicious.