What is the precise meaning of anatta? This is focusing in on one specific aspect of what was asked here: What are the three marks of existence?

I often hear "anatta" explained as being an expression of the non-existence of the self. However, various places add an adjective first. For example, I've heard it said that anatta refers to the non-existence of a "permanent" self, or a "material" self, or a "separate" self (or some combination of those and others).

It seems clear to me that it cannot mean the non-existence of a self per se, because such a statement -- "There is no self" -- would be auto-refuting.

So exactly what is meant by anatta?
Anatta is used in an adjectival sense, as a characteristic of something (all things including nibbana, in fact). So it doesn't have anything to do directly with a philosophy that "there is no self". This doesn't by any means suggest that the Buddha encouraged belief in a self. The best refutation of such an idea that I know of is in the Alagaddupama Sutta (MN 22):


  23.“Bhikkhus, you may well cling to that doctrine of self that would not arouse sorrow, lamentation, pain, grief, and despair in one who
  clings to it.262 "" But do you see any such doctrine of self,
  bhikkhus?”—“No, venerable sir.”—“Good, bhikkhus. I too do not see any
  doctrine of self that would not arouse sorrow, lamentation, pain,
  grief, and despair in one who clings to it.
  
  ...
  
  25.“Bhikkhus, there being a self, would there be for me what belongs to a self?” —“Yes, venerable sir.”—“Or, there being what belongs
  to a self, would there be for me a self?”—“Yes, venerable
  sir.”—“Bhikkhus, since a self and what belongs to a self are not
  apprehended as true and established, then this standpoint for views,
  namely, ‘That which is the self is the world; after death I shall be
  permanent, everlasting, eternal, not subject to change; I shall endure
  as long as eternity’—would it not be an utterly and completely foolish
  teaching?”
  
  “What else could it be, venerable sir, but an utterly and completely
  foolish teaching?”
  
  (Bodhi, Trans)


So what does anatta mean, then? According to the Visuddhimagga, it means specifically that the item in question has no core (asāraka), as per the third stage of insight knowledge:


  ‘Materiality, whether past, future, or present, is impermanent in the sense of destruction, painful in the sense of terror, not-self in the sense of having no core,’ is knowledge of comprehension. 
  
  Path of Purification, XX.6


The explanation of having no core means having none of the four kinds of self:


Sāmi attā - controlling self
Nivāsī attā - continuous self 
Kāraka attā - active agent self 
Vedaka attā - experiencing self 


These four types of self are explained in the Mahasi Sayadaw's discourse on the anatta-lakkhana-sutta. Briefly, according to the Visuddhimagga:


  In the sense of having no core because of the absence of any core of
  self conceived as a self, an abider, a doer, an experiencer, one who
  is his own master; for what is impermanent is painful (S III 82), and
  it is impossible to escape the impermanence, or the rise and fall and
  oppression, of self, so how could it have the state of a doer, and so
  on? Hence it is said, “Bhikkhus, were materiality self, it would not
  lead to affliction” (S III 66), and so on. So this is also not-self in
  the sense of having no core.
  
  -- Path of Purification XX.16


The Visuddhimagga goes on to give five characteristics of the five aggregates that make them each non-self (the entire list of 40 characteristics is worth reading, but the rest have to do with the other two main characteristics):


  
  as alien because of inability to have mastery exercised over them, and
  because of intractability;
  as empty because of their emptiness of the lastingness, beauty,
  pleasure and self that are conceived about them; 
  as vain  because of
  their emptiness, or because of their triviality; for what is trivial
  is called “vain” in the world; 
  as void because devoid of the state of
  being an owner, abider, doer, experiencer, director; as not-self
  because of itself having no owner, etc.
  as not-self because of itself having no owner, etc.;
  
  
  -- Path of Purification XX.19


Practically speaking, what does all this mean? Not a whole lot, since the point is not to understand intellectually the concept of non-self, but rather to experience it. From a point of view of experience, non-self is understood through observing that the five aggregates (i.e. the aspects of experience) conform to the explanations above. This understandling comes naturally through objective observation of the four foundations of mindfulness, so it's not really something to ponder over or doubt about.