Bursting is not a goal in itself - it is just a side-effect of using cheap and simple latex or rubber balloons.

You fill the balloon so it has slightly positive buoyancy. As it rises, it encounters lower outside air pressure, and expands to match it. Eventually it will rise to an altitude where the pressure is so low, that the latex is stretched beyond its limits and bursts. Typically at about 40km and about 100x expansion in volume.

To prevent bursting, you need a more complex material that will expand many times at first (to allow the balloon to rise to operating height), but will then stop expanding without breaking.