Maybe software running on your PC to capture everything you type, a voice transcriber that filters out your voice specifically and records that, and you've got a dataset that covers a lot of who you are.
Fine tune a model on that and boom, you're "immortal", and as LLMs get better and better, the fidelity of "you" gets better and better.
No downsides or fun for yourself, the main use I could see for it would be for something like being able to "talk" to your great great great grandpa one day.
I suppose that is the one retaining upside for the other person when it comes to immortality. But for the individual, none of the immortality fun remains. You aren't meeting or talking to them as you would in "normal" immortality, you are still dead and don't know if they even exist, you don't even get to ensure the right things are passed down.