[Falik]

But everyone has this sense of "me".

No. Not everyone

Enlightened minds do not have a sense of "me".

I would go so far as to say that it it will always be there for everyone, regardless of what they do or don't do, or how "enlightened" they are.

No. Your view is definitely incorrect from a Buddhist perspective. From a scientific perspective, your view is mere unverified unproven speculation.

They can spend talk indefinitely about how there is no "I" or "me", but at the end of the day, this sense of "I" will always remain. It will remain even for someone who has dedicated their life to dispelling the illusion of it.

Again, your view here is mere speculation. But you are asserting it as though it is fact.

Memory is not what is experiencing experience.

I did not say memory is experiencing. I said memory creates the sense of continuity.

And even if you lost all your past memories, you would still have the sense of "I".

The sense of "I"would be a 'new becoming' (ponobhavika).

...you just wouldn't have a story or a history to go with it.

Yes! Well spoken. You have agreed with me in this last point and answered your own question about a "stream" of experience.

Yes! You just wouldn't have a story or a history to go with it. Yes! The mind would not have a sense of continuity.

There is no stream of experience. There is no movie or DVD that can be played back, again and again. There is just momentary or present experience. The illusion of a "stream" is cause by selective and imperfect memory, as I said in my original reply to you.

Regards