The only part of the flesh which is "you" is the nervous system. Everything else can be hacked off or replaced without a loss of identity.
You could imagine a Ship of Theseus style replacement of all the neurons in your head with artificial ones. At what point would you stop being you?
Seeing as at least one episode of Star Trek dealt with such a thing, I wonder if any ever dealt with the possibility of a single individual with access to a transporter creating a continuous set of copies from the original scan, perhaps saving at intervals and allowing the instantiation of variously youthful or experienced copies to be made. The divergences over time in such a branching consciousness would make for an interesting sci-fi thought experiment, I think.