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You can use 1/4 cup applesauce or 1/4 cup mashed banana in place of an egg. I have done it many times. If using the banana you will get a slight banana flavor.
You can also use a few ounces of plain yogurt. I just eyeball it - 3 Tbs. maybe... greek maybe better.
I tablespoon of ground flax seed + 3 tablespoons of water for each egg.
Mayonaise = eggs + oil. There are whole recipes for cakes using this in leu of eggs. Google "mayonnaise cake" or "miracle whip cake" for proportions. It makes an incredibly moist cake.
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