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.