I make one from the Fanny Farmer cookbook and it's my favorite. Moist, surprisingly, wihtout any fats. It's not a "fat free cake" (that is, one that tries to imitate a cake with fats, but for dietary reasons eliminates the fat) but it's just that way.
Keep in mind that oil gives cake an illusion of moistness but it really ends up feeling greasy on the tongue and actually i think i remember reading that "moistness" is as much from the sugar and the water-keeping properties of the ingredients (bananas, to wit) as the fats.
Eggs are deceiving. Think of a hard boiled egg. Is it moist??? no! dry!!! When the egg cooks it's not moist. Keep that in mind. Though the egg yolk does have fat, and sometimes i get a nicer texture eliminating one white and putting in an extra yolk (e.g. when a recipe calls for two eggs, i put one whole and two yolks) but that's not necessary in this recipe below.
I've tried many others that have lots of additions making them seem like they'd be better, moister, etc. but this is my favorite and so simple.
3 ripe bananas, mashed
2 eggs
2 cups flour
3/4 cup sugar
1 tsp salt
1 tsp baking soda
1/2 cup coarsely chopped walnuts (i use raisins instead) optional
preheat oven to 350. Grease a loaf pan.
Mix bananas and eggs in a large bowl.
Stir in flour, sugar, salt and soda. Add nuts and/or raisins and stir just to blend.
Pour into pan and bake 1 hour.
Remove from pan to rack
serve warm or cooled.
Then, if you want a banana CAKE, you should try Rose Beranbaum's cake bible banana cake. It's truly amazing.