What is "Self"?

A famous tale from the old Khandogya Upanishad

A GOD and a demon went to learn about the Self from a sage. They studied with him for a long time. At last the sage told them, "You yourselves are the being you are seeking."
Both of them thought that their bodies were the Self. The demon went back to his people quite satisfied and said, "I have learnt everything that was to be learnt: eat, drink, and be merry; we are the Self; there is nothing beyond us."
The demon was ignorant by nature; so he never inquired any further, but was perfectly contented with the idea that he was God and that by the Self was meant the body.
The god thought at first, "I, this body, am Brahman; so let me keep it strong and healthy, and well dressed, and give it all sorts of enjoyments."
But soon he found out that that could not be the meaning of the sage, there must be something else to the instruction. So he came back and said: "Did you teach me that this body was the Self? If so, I see that all bodies die; but the Self should not die."
The sage said: "You are that."
Then the god thought that the vital forces which work the body were what he meant by the Self. But after a time he found that if he ate, these vital forces remained strong, but if he starved, they became weak. The god then went back to the sage and said, "Do you mean that the vital forces are the Self?"
The sage said: "You are that."
The god returned home once more, thinking that it was the mind, perhaps, that was the Self. But in a short while he saw that his thoughts were so many and diverse - now good, again bad; the mind was too changeable to be the Self. He went back to the sage and said: "I don't think that the mind is the Self. Did you mean that?"
"No," replied the sage; "you are that."
The god went home and at last found the true Self, beyond all thought: It was one and without birth or death, called endless, omniscient, and omnipotent Being - not body or the mind, but beyond them and yet manifesting through these vehicles. [Via 588 - taken from Khandogya Upanishad, 8.7-15]


Now, I'm still looking for "that", the true Self.