Suppose that rather than copying my brain, I adjoined it to some external computer in a kind of reverse-Ebborian act; electrically connecting my synapses to a big block of computrons that I can consciously perform I/O to. Over the course of life and improved tech, that block expands until, as a percentage, most of my thought processes are going on in the machine-part of me. Eventually my meat brain dies -- but the silicon part of me lives on.

This is very similar to the premise of Greg Egan's short story, "Learning to be me".