Fanny Farmer Fresh Banana Cake Recipe
Since this is a couple of DU iterations back, thought I'd bring this link into the current one. I just made it again yesterday and it was as fabulous as always. I didn't have enough ingredients on hand to double it this time, so to make it as a 13x9 instead of a 9x9, I just halved them all again. (Doubling creates both a 13x9 and a 9x9 cake.)
http://www.democraticunderground.com/discuss/duboard.php?az=view_all&address=236x44258
And don't forget the nutmeg (if you like nutmeg, that is!) I left it off the original ingredient list, because it isn't a part of the original recipe, but yesterday I used 1t. or so, IIRC (no promises on that.) 
= new reply since forum marked as read
Thanks for the recipe; wonder if it can be adapted for no-flour types, like me.
I wanted to tell you how happy I was to see your post on the daily thread the other day. I recall many from the past where, on a Saturday or Sunday a.m., you would cook up a storm! It was always fun to read what was happening in your kitchen.
Cher
Thank you for your sweet welcome back. So good to see you, too. 
As to adapting this cake for no flour, I'm not sure how possible that would be. This batter comes out pretty thick, still totally pourable and spreadable, but definitely not thin. Even so, the bananas add quite a bit of liquid content when beaten in and it comes out very moist, light and airy (as opposed to banana bread) where a main flavor component such as cocoa for a flourless chocolate cake, for instance, does not. That's just my theory off the top of my head, though.
That being said, if you can find a similar flourless recipe that may be worth a try. I'd hate to waste all the ingredients to try to adapt this one and have it totally fail to set, though. (I just need to be very frugal these days and can't take those sort of risks.)
I did a search for flourless banana cakes, didn't look at any of them so not sure how good they might be, but a good cook such as yourself could probably read through them to figure out which might be best, if not also any reviews there may be.
https://www.google.com/search?q=flourless+banana+cake&ie=utf-8&oe=utf-8
Wish I could have been more help, I'm just thinking it may not be doable.
