Ah, but we have two sets of teeth. The ones we have when kids fall out and are replaced by larger new ones. Some animals apparantly do have one set that grows endlessly and is ground down throughout their lives. As a side issue, I hear that sheep's teeth are ground down so much by chewing that they occasionally have to have dentures fitted to allow them to eat. These are called 'ewe-splints'. I am told this is true - honestly.
I know that the adult teeth are formed in the gums, beneath the milk teeth. If for some reason they do not develop you'll keep your milk teeth (which has happened to me with a couple of teeth). So it might appear that they are growing when they push through, but I don't believe that they actually do.