The most we ever had was 5 stacked on top of each other. 5 was a good amount, but as an average height woman, I really needed a step stool to get into the top cages. We have them elevated off the concrete floor with cinder blocks, so the stack was about 6' high. Never had any issues with it. Those cages are very light and very strong.

Now if you were thinking of putting like tons of slate tiles or something in them... I might reconsider.