This assumes that subjective identity is the same thing as personal identity in the philosophical sense, which presumes an answer to the question. Many philosophers (including me at times) have fallen into that trap and it is a reasonable posistion so I won't nitpick that too much.
Another possible way to get rid of personal identity, hypothetically speaking, would be "beings" that had no continuity of anything- psyche, form, consciousness, body, or even genes- over any period of time we would consider significant. Such beings would have an identity at a given point but said identity would cease to be in less time than we could blink.
This assumes that subjective identity is the same thing as personal identity in the philosophical sense, which presumes an answer to the question.
Not being a philosopher, I don't know the difference.
Although Elizier has dealt with personal identity questions (in terms of ruling out the body theory), he has not actually, as far as I know, "solved" the problem of Personal Identity as it is understood in philosophy. Nor, as far as I know, has any thinker (Robin Hanson, Yvain, etc) broadly in the same school of thought.
Why do I think it worth solving? One- Lesswrong has a tradition of trying to solve all of philosophy through thinking better than philosophers do. Even when I don't agree with it, the result is often enlightening. Two- What counts as 'same person' could easily have significant implications for large numbers of ethical dilemnas, and thus for Lesswrongian ethics.
Three- most importantly of all, the correct theory has practical implications for cryonics. I don't know enough to assert any theory as actually true, but if, say, Identity as Continuity of Form rather than of Matter were the true theory it would mean that preserving only the mental data would not be enough. What kind of preservation is necessary also varies somewhat- the difference in requirement based on a Continuity of Consciousness v.s a Continuity of Psyche theory, for example should be obvious.
I'm curious what people here think. What is the correct answer? No-self theory? Psyche theory? Derek Parfit's theory in some manner? Or if there is a correct way to dissolve the question, what is that correct way?