Two ideas that I've seen on Instructables (but never tried personally): Crockpot Cake https://www.instructables.com/id/Crock-Pot-Molten-Lava-Cake/ Microwave Cake https://www.instructables.com/id/5-minute-Chocolate-Cake/
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Lava Cake in less than 15 minutes What Cake in Less than 15 minutes without EVER having to turn the oven on. Cake can't get any easier than or faster than this!! This cake is super moist and so tender. Plus I LOVE the "lava". :) What you Need: 1 box Cake mix (and everything needed for the box mix) 1 can frosting What to do: Make cake batter according to box directions in a Microwavable Dish, like Our Pampered Chef Deep Dish Baker. Using an Ice Cream Scoop, (Like our awesome Medium Scoop) Scoop out half of frosting from the can, and plop into center of cake batter. . Microwave for 12 minutes.Let cool about 2 minutes Invert a plate ontop of dish and flip cake out onto plate.
Bake with a pressure cooker... we make a REALLY nice syrup sponge in the pressure cooker Alternativly, during the war they used a wooden box filled with straw to cook things, not sure how it works but i have seen it done once or twice... that might work
Here's an Instructable on using a pot as an oven to bake cakes.https://www.instructables.com/id/Homemade-Oven-for-Pizza-and-Cakes/
look up on google HOW TO BAKE A CAKE IN A MICROWAVEor just use a friends ovenwe did my first idea on a friends birthday and it turned out well
Box-type solar cooker. You can make them pretty easily and all you need is a sunny day.
There are cakes where the base is made of grinded coockies together with butter and it just needs to set in the fridge (so no backing involved)
Easy Bake Oven used a lightbulb...
A friend's wife makes a Dutch oven cake whenever they have a barbecue. She makes a chocolate cake that is scrumptious.
Sponge cakes can be baked in a microwave. Make a no bake cake with ice cream and crushed cookies.
Assuming that you have a heat source such as gas ring/ electric ring or woodstove. . . We used to cook our pies in a big le crueset casserole pot (cast iron), on a gas ring. The pies were in foil dishes, raised from the floor of the pot on a little piece of metal mesh. I can't see why this wouldn't work for a cake. Cast iron pots are quite cheap at my local amenities tip.
My mother-in-law used to use this method with great success:http://www.suite101.com/content/cooking-cake-on-the-grill-a136001