I don't have a real example, obviously, but there are a few possible ways this could be done:
* Show it ends in a loop other than 4,2,1, as mentioned
* Show that it has a repeating pattern of operations that makes it keep growing. As an example, if you can show that you'll get x -> ... -> 3 x -> ... -> 9 x -> ... -> 27 x and so on forever, then you know x is a counterexample to the conjecture.
* Show it has some property which won't go away under the steps of the Collatz conjecture, and show that 4,2,1 don't have this property.
* Show that it has some property which will make it (after an unknown number of steps) go to a larger number with the same property.
As a (trivial) example, consider "if the number is odd then add 2, if the number is even but larger than 0 then subtract 2, if it's 0 then don't change it". If you start with 1 it's immediately obvious that you will never reach 0 with it. You can formally prove this by showing that 1 is odd, 0 is even, and the transformation never changes an odd number to an even number.