Since the most salient feature of a new toy is generally its novelty, you can make a perfectly adequate play stove by flipping over a cardboard box and drawing some burners on it. This stove is insanely overbuilt; I was mostly looking for an excuse to try out making "plywood" out of cardboard.
So, on to the cardboard laminate. A useful fact from the world of mechanics of solids is that the flexural rigidity of a plate increases with the cube of the thickness. In layman's terms this means that if you double the thickness, you make it eight times stronger. Corrugated cardboard already takes advantage of this fact by weaving a sheet of kraft paper back and forth between two flat sheets of kraft paper, making it much stiffer than it would be if the layers were all flat.
By gluing up many pieces of corrugated cardboard, we can make cardboard panels that are an inch or more thick and as large as we want. At 1" thickness, these panels are already quite strong; they can support several hundred pounds across a 24" span.
Not only are the panels made from recycled materials, they are themselves recyclable, so when all the fun has been exhausted, the stove can go out with the rest of the cardboard recycling.
Step 1: Save Your Cardboard Boxes
Once you have some "big" pieces, let the usual assortment of smaller shipping boxes accumulate until you have a good supply of raw materials, or your spouse starts yelling at you. Break down the boxes and remove any staples or overlapping segments---it's important that all of the pieces in a given layer be the same thickness. If you have boxes of varying thicknesses, sort your boxes by thickness.
The other things you'll need are:
* a gallon container of Elmer's glue
* a plastic drop cloth
* a utility knife
* some kind of saw. (I used a table saw, but a hand saw would work fine and would probably be less scary.)
* kraft paper packing tape
For the play stove, you also will need:
* 4' of bungee /shock cord
* some stove knobs, handles, black paper for the burners etc.