[cypher]

I agree that joining the cages side by side would be far better but if you do have to go vertical have you thought about how much floor space would be left after cutting the access holes? You would probably end up with too little really useable space on each level & you would also have to think about the possibility of falls even with ramps in place.
Whatever you choose to do you will have to use non toxic glue, even if there's no chance of it being chewed (& you can never really be sure of that!) some glues give off dangerous fumes.