What does Buddhism say about the existence of a self?

The 'self' is a temporary & changing 'idea' or 'assumption' the ignorant mind makes in relation to different experiences. Each separate idea of 'self' is formed by the process of becoming.

For example, a young child first develops ideas of self, such as "my mother; my father" & "I like this food". As the child gets older, it develops ideas of self such as: "I am American"; "I am Buddhist"; "I do not understand anatta"; " I disgree with anatta", etc. Before any experience of Buddhist teachings occurred, there was never the self-view of: "I do not understand anatta". The idea "I don't understand anatta" only occurred after having a sense experience of Buddhist literature.

All things are not a 'self' because any idea of 'self' is an 'assumption' and is 'suffering'. Thus, the scriptures have the saying: 'there is no self to be found; all there is is suffering arising & suffering ceasing'. This is because the arising of self-view is considered to be, in reality, the arising of suffering.

This is why there is no real self or even a temporary self because what unenlightened people call a 'self', the enlightened mind views as 'suffering'.

When enlightened minds use the words "I" and "mine", they regard such words as mere worldly social conventions rather than a temporary self.

People will refer to the Ananda Sutta about Vacchagotta to answer your question but this is invalid since here the Buddha refused to comment on the matter of self because Vacchagotta was confused & bewildered and incapable of understanding.

Some quotes below:

There is the case where an uninstructed, run-of-the-mill person — who has no regard for noble ones, is not well-versed or disciplined in their Dhamma; who has no regard for men of integrity, is not well-versed or disciplined in their Dhamma — assumes form to be the self. That assumption is a fabrication. Now what is the cause, what is the origination, what is the birth, what is the coming-into-existence of that fabrication? To an uninstructed, run-of-the-mill person, touched by that which is felt born of contact with ignorance, craving arises. That fabrication is born of that. And that fabrication is inconstant, fabricated, dependently co-arisen. That craving... That feeling... That contact... That ignorance is inconstant, fabricated, dependently co-arisen.

Parileyyaka Sutta

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Why now do you assume 'a being'? Mara, have you grasped a view? This is a heap of sheer constructions: Here no being is found.

Just as, with an assemblage of parts, The word 'chariot' is used, So, when the aggregates are present, There's the convention 'a being.'

It's only suffering that comes to be, Suffering that stands and falls away. Nothing but suffering comes to be, Nothing but suffering ceases.

Vajira Sutta

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Bonds are gone for him without conceits, All delusion's chains are cast aside: Truly wise, he's gone beyond such thoughts. That monk still might use such words as "I," Still perchance might say: "They call this mine." Well aware of common worldly speech, He would speak conforming to such use.

Araha.m Sutta

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"And if I — being asked by Vacchagotta the wanderer if there is no self — were to answer that there is no self, the bewildered Vacchagotta would become even more bewildered: 'Does the self I used to have now not exist?'"

Ananda Sutta SN 44

So why is 'self' viewed as 'suffering' or as a 'disease' rather than viewed as a 'self', even as a 'temporary self'?

This is because each arising & creation of 'self' is based in craving and craving is suffering.

For example, once a child's mind is developed enough to manufacturer ideas of 'self', it manufacturers ideas of 'self' based on what it craves. These cravings are suffering or frustrations.

For example, the five aggregates of the child crave for food due to hunger, so the child manufacturers the thought: "I want food" which leads to habitual thoughts: "I like this food".

Or the child feels the fear & terror of loneliness and manufacturers the thought: "I want my mother", which leads to habitual thoughts: "I love my mother".

When the child grows older, it craves to succeed at school or sport, due to the fear of social failure. It manufactures the ideas: "I must succeed" or "I am good at sport". The child now views itself as "I am sportsman", "I am a student".

Then later the young adult has sexual craving, which is suffering & frustration. The young adult, pushed by the suffering of sexual craving, obtains a partner and develops the self-view: "I am a boyfriend"; "This is my girlfriend", "I am a husband", "This is my wife".

This is why each idea of self is considered to be, in reality, merely suffering. All ideas of 'self' are produced by some form of craving, however gross or subtle.