One instance of the failure to self-diagnose the fallacy of composition that I have come across in a real conversation is this: Person X that I am conversing with says "Anything you observe in the universe has a cause/has something else outside of it/there was a time before which the thing existed. There has to be a cause for the universe/there is something else outside of it/there was a time before which the universe existed." This fallacy is very common, but this example is one of the classic ones.
Check out these two portentous videos: Reassembling modular robot, Replicating robot
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