no
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they didn't, it just kept on going and never ever stop.
However, big a number is that you can think of , then by adding '1' it becomes larger/bigger. Numbers are infinite.
Numbers go on for infinity (for ever , so basically they never stop.) therefore there aren't any "largest numbers". I Know this is probably not what you wanted to hear but i hope i have been of help. :)
There are no two "last" composite numbers. Just as with prime numbers, and all numbers, they go on for ever and ever and ever and ever ... ... .
Numbers never stop, because they go on forever. So really, when all the numbers stop, we'll all be dead.