Closed Cell Polyurethane Foam
There have been attempts to create flat free tires of some sort or another since the early 1920's. Flat free tires have come in a variety of mediums from solid rubber to an "open-cell" foam similar to what you find in Nerf® football. Solid rubber tires are quite heavy and provide very little, if any shock absorbency, while open cell foam tires are easily abrased and tend to absorb moisture.
Another attempt to make a tire flat free is to pour material into the inner tube of the tire prior to inflating it. This method did not keep the tire from going flat, but merely delayed the deflation of the tire once punctured.