Hi L.,
I've got two of the most delicious egg-free cakes that I make all the time. They're so good, you can even eat without frosting. The first is a chocolate cake:
3 C. flour, 1 1/2 C. sugar, 6 TBSP cocoa powder, 2 tsp. baking soda, 1 tsp. baking powder, 1 tsp. salt, 2 tsp. vanilla, 2 TBSP vinegar, 2 C. water, 2/3 C. veg. oil. Put all dry stuff into bowl and mix. Make a well in center and add wet stuff all at once. Mix thoroughly and put in ungreased 9x13 pan. Bake at 350 for 30-35 minutes. Done when you can tap lightly on top and it rises back up. This is so easy and so good.
The second recipe is for applesauce cake: 2 C. flour, 1 C. sugar, 1 tsp. salt, 1 tsp. cinnamon, 1/2 tsp. nutmeg, 1/4 tsp. cloves (or allspice if you don't have cloves), 2 tsp. baking soda, 1/2 C. melted butter, 1 1/2 to 2 C. applesauce, 1 C. raisins and/or 1 C. chopped nuts (both optional). Throw all in a bowl and mix thoroughly. Bake in 9x13 pan 35 minutes at 350. This cake is really good plain, or with cream cheese frosting. You will never miss the eggs in either of these cakes. Happy baking!