Anatta does NOT have to mean "No self". Anatta = "not Atman" where Atman is a Hindu metaphysical being not a modern western concept of "self".
The earliest use of word "Ātman" in Indian texts is found in the Rig Veda (RV X. 97. 11). Yāska, commenting on this Rigvedic verse, accepts the following meanings of Ātman: the pervading principle, the organism in which other elements are united and the ultimate sentient principle. Yajnavalkya (c. 9th century BCE), in the Brihadaranyaka Upanishad, uses the word to indicate that in which everything exists, which is of the highest value, which permeates everything, which is the essence of all, bliss and beyond description. link
Within Advaita Vedanta philosophy the Atman is the universal life-principle, the animator of all organisms.
In SN44.10 there is interesting passage:
If I — being asked by Vacchagotta the wanderer if there is [no atta, natthattā] — were to answer that there is [no atta, natthattā] , that would be conforming with those brahmans & contemplatives who are exponents of annihilationism
Since, as I understand it, Annihilationists do believe in empiric self that dies during death, I think this sutta quote suggests that atta means something more than mere empiric self.
When we examine some of Hindu Philosophical teachings on Atman, it doesn't really sound like what we, in English, call "self" in ordinary, non-philosophic usage. In the suttas the Buddha seems to perfectly well to refute Atman by showing anicca and dukkha of everything knowable.
I wonder If someone would ask the Buddha, "why can't self be anicca, and dukkha? what is wrong with that?"
Concept of self, as in empiric, and ordinary usage doesn't need to mean the metaphysical entity that never changes and is always happy. In Christianity, soul can suffer in hell, and yet it doesn't refute its existence. So even Christian idea of soul would not fit Atman as unchanging and always happy first principle. So it seems that Buddha was rejecting something else.
Word for self in pali is sayaṃ. Not-Self is asayaṃ (a+sayaṃ).
sayaṃkata= done by oneself
sayṃvara= self choice.
asayaṃvasī = not under one's control.
What is the difference between Atta, and sayaṃ?