The only other way for a number string not to loop is to have an infinite chain of numbers which never repeat.  (One way this could happen would be for each number in the chain to be bigger than the one before it, but it is far more likely that one of these chains would meander back and forth while generally increasing).  It is impossible to demonstrate one of these, for obvious reasons, but that doesn't mean that they don't exist. My unprovable guess is that none exists, because at each step in the chain the next number would need to be not only one which is not earlier in the chain, but also not one of the infinite number which can be reached by working backwards from 16.