I've seen some amazingly fat tires on some pretty skinny rims. One issue that no one has mentioned is clearance. You will get to some point with fatter and fatter tires where they may not clear - either height or width - the fork crown, the fork stays, or, in the rear, the seat stays. With road bikes, the clearances are just tiny at times. 2mm is not unusual.
Mounting problems seem to come with some combinations of certain tires and certain rims. It's pretty much hit or miss. Most are not a problem and you should be able to remove with just a lever and mount with your hands. For really stubborn mounts the KoolStop TireJack is an awesome tool. Plus, it costs less than $10.