Did the Buddha leave room for the possibility of a self?

"Did the Buddha leave room for the possibility of a self?" Well, let's let THE MAN himself answer your question:

DN 2.157 Therefore Ananda, stay as those who have their True Self as the illumination, as those who have their True Self as supreme refuge, as those who have no other as the refuge; as those who have the true law Dharma as the illumination, as those who have the Dharma as refuge, as those who have no other refuge.

KN 3.78 And whoever, Ananda, either now or after my end will stay as those who have the True Self as the illumination, as those who have True Self as refuge, as those who have no other as the refuge…they among my bhikkhus shall reach the peak of immortality, provided they are desirous of training their True Self.

SN 3.174 Like a surge of the great ocean, so also will birth and death roll over you like a surge. Therefore, do make your True Self the supreme illumination, since there is no other refuge anywhere to be found for you. 4. My life is fully ripe, my life is at an end, I shall depart leaving you, I have made a supreme refuge for the True Self.

DN 2.24 Do make your True Self the illumination, strive fast, be wise. Having removed all stain, flawless, you will come to the divine Ariyan land.

AN 3.21 The phenomenal world all round is devoid of true essence, the four quarters are quaking. Desirous of an abode for the True Self, I saw none occupied.

KN 5.77 Is there by any chance any other dearer to you Mallika than the True Self? No Lord, there is not by any chance that which is dearer to me than the True Self.

AN 2.79 Going around all quarters with the mind. Not a thing was found dearer to me than the True Self. In this way the True Self of every one is dear to others.

MN 3.119 There is no love comparable to that of the True Self.

KN 5.38 One should not impair the good of the True Self, for the sake of the good of others, however great. Having ascertained the good of the True Self, let him be ever intent on it!

SN 1.109 One watches zealously over that which he holds dearest. This should apply to the True Self better than to anything else: If a man were to think the True Self dear, he would guard it most well guarded. The wise man should be watching in every one of the three watches of the night.

MN 4.83. And what does it mean to guard the True Self? Lord, while I was meditating in solitude, there arose in my mind the following thoughts. By whom is the True Self guarded, by whom is the True Self is not guarded? Then it occurred to me, whoever misbehaves by action, by word, or by thought, are those by whom the True Self is not guarded. Even if they were guarded by a troop of elephants, or horses, or of chariots, or of infantrymen, even so their True Self most dear would not be guarded by them.

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AN 2.234. Bhikkhus, I shall keep the True Self safe, this means that the stations of antecedent-recollectiveness of Samadhi must be dwelt upon intently.

SN 3.216. The True Self, the dearest thing for man, becomes an absolute value, which has to be preserved by all means and in preference to everything else: What should a man desirous of his own good never give up? What should a mortal man never surrender? Man should never give up the True Self most dear, he should never surrender the True Self.

AN 2.119. Him for whom the True Self is not enough, who procures for the True Self the taste of all sensual pleasures, even if the whole world were his, he would not obtain true bliss.

DN 3.218. Lord, while I was meditating in solitude, there arose in my mind the following thoughts. For whom is the True Self a dear friend, for whom is the True Self a hateful enemy? Then it occurred to me, whoever misbehaves by action, by letter of the law or by thought, are those for whom the True Self is a hateful enemy. Even if they were to say, the True Self is our dear friend, even so the True Self would be to them a hateful enemy. Why so? Whatever one who hates would do to the one he hates, that is what they themselves does unto their True Self. That is why the True Self is a hateful enemy to them. Whoever behaves properly by action, by letter of the law, or by thought, are those for whom their True Self is a dear friend.

DN 3.29. If he would recognize the True Self as dearest friend, he would not associate it with evil. 18. Then what do you think youngsters, what is the best thing for you, that you go in search of a woman or that your go in search of the True Self most dear? This Lord, is the best for us, that we go in search of the True Self!

SN 4.67. Bhikkhus, wise and developing a boundless penetration of antecedent recollection. A fivefold knowledge arises in their True Self in the case of those who, wise and immersed in antecedent recollection of the source develops a boundless penetration of antecedentness. What fivefold knowledge? This antecedent recollectiveness is pleasant at present and will yield a pleasant karmic result in the future, such knowledge arises in their True Self. This antecedent recollectiveness is noble, entirely spiritual and otherworldly, such knowledge arises in their True Self; this antecedent recollectiveness of the source is not practiced by the unworthy man. This antecedent recollectiveness is peaceful, most exquisite, obtained by the peaceful man, attained by means of mental fixation, not subject to the blame of the Sankaras. I too being in antecedent recollection of supreme beforeness in connection with the source enter into it and in antecedent recollection I emerge from it. Such profound knowledge arises in their True Self.

KN 5.18. A Buddha has arisen in the world, the doctrine of the Buddhas is at present being taught. The True Self can be saved by a man desirous of this doctrine.

KN 1.118 Whoever looks for the happiness of the True Self, should pull out the mortal dart of the True Self.

DN 2.281. Whose faith and wisdom are always properly fitted to the yoke, Supreme alert vigilance is the pole, mind is the yoke-straps, antecedent recollectiveness of the source is the guard and the Charioteer. The chariot having all the accessories of good faring, otherworldly knowledge as the axle, vigilant energy as the wheels. Equanimity is the fitting peg for the axle, desireless for fain of this world is the protective board. Excellent equanimity, deathlessness, and seclusion being the weapons, endurance the leather armor, it proceeds towards utter security. Such is the unsurpassed Brahman chariot produced in the True Self.

DN 3.28` Even as a deviating cart out of control, unrestrained, unmastered, destroys both the cart and the rider, in the same way the reckless fool, like a deviating cart destroys his True Self in hell, destroys the True Self in animal rebirth, destroys the True Self in the realm of wandering ghosts and spirits, destroys the True Self in the world of men, destroys the True Self in the world of gods. AN 2.28. And how is one a knower of the True Self? Herein bhikkhus, a bhikkhu knows the True Self. Just this much am I as regards faith, virtue, learning, disembodiment, wisdom, intelligence.

SN 2.97 Leaving aside the five hindrances for the obtainment of utter security. Taking up the mirror of Dharma for the knowledge and vision of the True Self, I observed the body both within and without, interiorly and exteriorly the body appeared to be empty.

MN 4.57 No Brahman ever claimed purity from any different source than the True Self. Either in things seen, heard, thought, or in observances. Unattached both to good and evil deeds, disclaiming whatever is obtained, he should be inactive in these observances. The ultimate purity which is the ideal of the enlightened man, here called a true Brahman, is a purity unaffected both by moral good and by moral evil, belonging to the plane superior to both, consisting in a condition that is reflected in the total absence of willful moral activities, in the absolute desirelessness to do evil and to obtain fruitless merit. This is the absolute isolation of the True Self which brings about liberation. The improvement caused by morality and moral practices is meant first of all to detach the True Self from what is evil, and this is mainly done by the counterpractice of goodness. This is not enough; any attachment of the True Self to whatever is not the True Self is itself wrong from the ultimate point of view. Moral good and the subsequent merit is not the True Self, even though it takes the True Self towards an ever more perfect detachment from worldly things. Finally the True Self has to be detached from morals, morality, merit and be freed with a freedom that is its very nature.

AN 2.281 Hence, let the wise man, discerning the welfare of the True Self, thoroughly investigate the Dharma, thus thereby he will be purified.

SN 2.27 I shall apprehend and perfect the True Self, having in mind the spiritual welfare of the True Self.

AN 4.76. Force the True Self by means of the True Self, control the True Self by means of the True Self. Being well guarded of the True Self, in antecedent recollectiveness, you shall bhikkhus dwell in supreme bliss.

DN 1.195 Leaving aside the way of darkness, the wise man should practice the way of light. Going from home to the homeless state, in solitude, where worldliness joys are difficult, there should he desire for the unexcelled bliss, setting aside sensory pleasures, possessing nothing. Let the wise man cleanse the True Self from the impurities of the mental goings on.

KN 3.28 Just as the goldsmith melts and removes the gross impurities of gold, then melts and removes average impurities, and melts and removes even the finest of impurities. Just so does the bhikkhu melts, removes, leaves aside, dispels, destroys, the impurities of his True Self.

AN 5.73 Not by heaping up firewood does the Brahman dream of purification. That is something external. Because, so the wise say, purity is not obtained by him who wishes to reach gain by means of external rituals. I, leaving aside the burning of wood, Brahman, make only destruction come unto those flames that are attached to by my True Self. With fire constantly burning, always with my True Self well composed, I that very Arahant, live my Brahman life. A shoulder yoke, Brahman is your conceit, anger is your smoke, your false words are the ashes. The tongue of the man is his sacrificial spoon, the heart his fire alter. The self well tamed is the fire. Dharma O' Brahman, is a lake with holy virtue as the bathing place, pure undefiled, praised by the good. Where the wise bathing, with their True Self disembodied, do so cross sweetly to that other shore.

AN 4.78 Where water, earth, fire, and air find no footing, there where the stars no longer shine, nor the sun, nor does the moon gleam; no darkness is found there. And when the mighty sage, that holy Brahman, has come to supreme knowledge by the True Self.

KN 2.118. One who has made a path by the True Self, he does so go unto complete retraction from aggregated being, having crossed all doubts. Leaving aside becoming and passing away. One who has lived the life, who has suppressed all rebirth, such a one is called a true bhikkhu.

KN 4.283. Sweetly within antecedent recollective penetration as regards the body, restrained within the six sensory spheres. The bhikkhu who is well composed would know the complete retraction from aggregated being of his True Self.

AN 1.81 There is monks, an unborn, an unoriginated, an unmade, and an unformed. If there were not monks, this unborn, unoriginated, unmade and unformed, there would be no way out for the born, the originated, the made and the formed.

SN 4.57 And I O' monks, who speak thus, and teach thus am accused wrongly, vainly, falsely, and inappropriately by some ascetics and Brahmins who say "A denier is the ascetic Gotama, he teaches the destruction, annihilation, and the perishing of the being that now exists". These ascetics wrongly, vainly, falsely, and inappropriately

Buddha said Sabbe Dhamma Anatta. Anatta means no self. There is no permanent , unchanging self. How much clear should Buddha have made it ? There is no scope of Self as far as Dhammas are concerned.

In Sravakayana Sutras he stated that any dependently originated phenomenon are not self and in Mahayana Sutras he makes it clear that the Tathagatagharba and Nirvana hold the qualities of Permanence, Bliss, Self and Purity.

SN 35.85 states the world is completely void/empty (sunnata) of self or anything belonging to self.

SN 5.10 and SN 12.15 state there is no temporary conditioned 'self' because, in reality, what is regarded as a temporary delusion or view of 'self' is really only 'suffering'.

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.

SN 5.10

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By & large, Kaccayana, this world is in bondage to attachments, clingings & biases. But one such as this does not get involved with or cling to these attachments, clingings, fixations of awareness, biases or obsessions; nor is he resolved on 'my self.' He has no uncertainty or doubt that only stress, when arising, is arising; only stress, when passing away, is passing away. SN 12.15

In the Pali canon, it seems there is no categorical denial of a permanent self by the Buddha.

The most common texts about the not-self (anatta) doctrine are taught in the form " [something] is not self", where [something] is a phenomenon that is cognized: forms, feelings, thoughts, perceptions, etc. For example:

“Feeling is impermanent…. Perception is impermanent…. Volitional formations are impermanent…. Consciousness is impermanent. What is impermanent is suffering. What is suffering is nonself. What is nonself should be seen as it really is with correct wisdom thus: ‘This is not mine, this I am not, this is not my self.’

-- SN 22.45

A generalization of this teaching is found in AN 3.136:

sabbe dhammā anattā'ti
all phenomena are non-self (Bodhi translation)

Where "phenomena" (dhammā) stands as a constituent of experience, something or anything that can be experienced.

Now, the Buddha has been asked point blank if there's a self, but he refused to answer:

Then the wanderer Vacchagotta went to the Blessed One and, on arrival, exchanged courteous greetings with him. After an exchange of friendly greetings & courtesies, he sat to one side. As he was sitting there he asked the Blessed One: "Now then, Venerable Gotama, is there a self?"

When this was said, the Blessed One was silent.

"Then is there no self?"

A second time, the Blessed One was silent.

Then Vacchagotta the wanderer got up from his seat and left.

-- SN 44.10

He then proceeds to explain why he did not answer (another explanation is given in SN 44.7):

"Ananda, if I — being asked by Vacchagotta the wanderer if there is a self — were to answer that there is a self, that would be conforming with those brahmans & contemplatives who are exponents of eternalism [the view that there is an eternal, unchanging soul]. 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]. If I — being asked by Vacchagotta the wanderer if there is a self — were to answer that there is a self, would that be in keeping with the arising of knowledge that all phenomena are not-self?"

— "No, lord."

"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?'"

Some people take that Vacchagotta episode, plus the lack of any categorical denial and a few other texts and try to create evidence that the Buddha actually taught a permanent self doctrine. But all of them are considered attempts of fitting a square into a circle hole -- usually they are not very convincing.

It's quite reasonable that If there was a permanent self that could be grasped, all the Buddha had to do was declare it and teach it. It would be much easier to understand then his doctrine of dependent origination and he wouldn't have to answer puzzling questions like "if there's no self, what reborns?".

To the best of my knowledge, the closest refutation for a doctrine of permanent self in the pali canon is the following sutta:

"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. 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."

-- MN 22

If this teaching holds true, then the only possibility for a permanent soul to exist is if it exists outside the reach of experience, outside the possibility of contact.

But if we can't reach it to assess if it exist or not, if we cannot come into contact with it, then either it doesn't exist, or it exists but doesn't seem to have any effect in our existence -- since it's completely isolated from our experience. Thus, even if no phenomenon is our permanent self, but such a self still exists somewhere else, certainly it is irrelevant for our life, and believing or not seems to do little to help in the path towards nirvana -- after all, that permanent soul does not affect us in anyway.

Buddha did not teach that things exist (that's one extreme) or that things do not exist (that's another extreme). Buddha teaches that things exist to some extent, in some context, relative to some frame of reference – and do not exist in other contexts. It's the same with the "self".