The Buddhist answer is 'self neither exists nor doesn't exist'. To say 'self exists' is to fall into the view of permanent existing subject, or 'eternalism'. To say that it doesn't exist is to fall into the error of 'nihilism'. So it is not a question with a simple yes or no answer. In fact the answer is often silence as the question is the subject for contemplation, not amenable to any easy answers.
Odsal wrote: Inquiring into the nature and existence of an experiencer is inquiring about the nature and existence of something that doesn't actually exist.
Ananda Sutta wrote: If I — being asked by Vacchagotta the wanderer if there is no self — were to answer that there is no self, that would be conforming with those brahmans & contemplatives who are exponents of annihilationism [the view that death is the annihilation of consciousness].
From Ananda Sutta
That is why I experience the elements of the cosmos; earth, fire, water, wind and space because that is what "I" am, not this abstract entity that is located behind my eyes, inside my head or brain or wherever we may conceive'
it is true that the subject cannot be objectified or be thought of as 'a mysterious entity'. But neither is it simply the sum of the elements. I think that is too close to materialism.
The Brahmins would say this is Atman, self. In my view, the Buddha was simply more consistent about what could and could not be 'named and known'. To name something as 'Atman' is to implicitly make it an object of cognition. The Upanisads have many passages saying 'about this Atman, nothing can be said'. But that doesn't stop them from talking about it at length. I think the Buddha was simply much more consistent about it. (This is discussed at length in Richard Gombrich What the Buddha Thought
I realize it is an unanswerable question. I mentioned that towards the end of my first post. I said something like: the only way the confusion is cleared up is through insight into the actual nature of mind and appearances. Because otherwise one would be engaging in discussion about a concept as if it were a real thing. Like chasing a mirage. Pondering the existence of unicorns and stuff.
I'm not a nihilist or materialist though, however my statement may sound. Reality clearly is not static, it has no starting point or end point. Empty appearances are never ultimately established as self-existing, they never actually come into existence in a fixed permanent way due to their being dependently originated and therefor cannot be said to ultimately cease. This is clearly seen in how appearances change according to conditions. So a view of non-existence is clearly not in accord with the fact that reality functions.
I didn't mean to suggest that subjective experience is simply the sum of the physical elements. What I mean is that what we define within our perception as objective, i.e.the elements, is not external or separate from what we define within our perception as subjective. Thus, it wouldn't make sense to hold the view "subjective experience is the sum of the elements" or hold the view that "the subjective is the creator of the objective".
The world view that fundamentally holds reality to be "Me" on this side and "everything else that is not me" on the other side is a mistaken view of reality, a non recognition of dependent origination. It is seeing the parts, but not seeing how they are interconnected and as a result taking them to be independently existing phenomena. I think a more useful question to ponder would be: "what is the cause of the arising of this dualistic mind set of subject and object?".
These are just my thoughts. I hope they are useful are at the least entertaining.