Depends on the stretchiness of the material. Even without a confined space, but in a theoretical magically fixed balloon tower, I’m going to say that they’d bounce forever. The reasoning being that at some point the bottom balloon would be fully stretched but since it can push against the second balloon, which is not yet fully stretched, it can make that stretch, so it wouldn’t necessarily burst. Continuing this feedback effect upwards it would mean the average balloon only needs to hold its own weight.