You would first have to prove that the number is actually a counterexample. I don't know what you mean by calculating the whole sequence, as it would be an infinite sequence. Proving a counterexample seems like a halting problem kind of problem.
But the question of a counterexample (eventhough it probably doesn't exist) is interesting. Is there a number for which we don't know weather it is a counterexample because it would take too long to compute? I feel like any single counterexample you'd propose would be proven wrong pretty quickly.
The problem is if my counter example is a counter example it diverges to infinity. (someone else proposed a proof that there are no cycles other than at 1, if this proof is wrong my counter example could be a cycle and thus "easy" to show). However if goes to infinity it is really hard to see how you show it doesn't eventually converge if you just went a little big longer.
Note that every counter example you propose is actually a sequence of counter examples. It would be interesting to examine the properties of whatever numbers that counter example has in common. Though this is an obvious thing that I suspect someone has already done to no effect.
