There are some who will claim that by stating the aggregates are not self, the Buddha left open the possibility of a self beyond the aggregates. Yet this feeble attempt at an argument falls apart immediately upon investigating the Buddha’s teachings with any depth.
To begin, it may be helpful to consider the context in which the Buddha spoke of self. At the time of the Buddha, the varied schools of religious and spiritual practice and philosophy veered to two extreme doctrines: Sassatavada and Ucchedavada, or eternalism and annihilationism, respectively. Whether the Brahmanic tradition that informed present-day Hinduism whose adherents invested their faith in the existence of an eternal Godhead described in the Vedas and Upaniṣads, or the materalistic schools of the time that denied the existence of any sort of afterlife whatsoever, both were guilty of extremism in their dogmatic approaches to metaphysics. In contrast, the Buddha, as is claimed on countless occasions, taught the “middle path,” the “middle way,” the “middle road” between these extremes: Majjhimā Patipadā.
In reference to the other religious teachers and philosophers of his time, the Buddha spoke the following discourse:
“Monks, whatever contemplatives or brahmans who assume in various ways when assuming a self, all assume the five clinging-aggregates, or a certain one of them.”
“Samanupassana Sutta: Assumptions” (SN 22.47)
Here the Buddha notes that when others affirm the self, they do so in a particular way. All point to one or some combination of the five aggregates: form (rūpa), feelings (vedanā), perceptions (saññā), mental formations (saṅkhāra), and consciousness (viññāṇa). After all, these are the aspects of our psychophysical makeup that are most easily mistaken for a self of some kind or another.
In the case of the Sassatavadins or eternalists, a tendency to identify self with various aspects of mind emerged, allowing for the existence of an eternal soul that outlives the body. This is reflected in early non-Buddhist, Brahmanic texts such as the following, in which ātman is identified with mind at various levels:
That Atman (self, soul) is indeed Brahman. It [Ātman] is also identified with the intellect, the Manas (mind), and the vital breath, with the eyes and ears, with earth, water, air, and ākāśa (sky), with fire and with what is other than fire, with desire and the absence of desire, with anger and the absence of anger, with righteousness and unrighteousness, with everything — it is identified, as is well known, with this (what is perceived) and with that (what is inferred).
—Brihadaranyaka Upanishad 4.4.5
This tendency to identify self with mind (among several others) is mistaken according to the contemplative framework within which the Buddha taught, which logically and straightforwardly laid out the following formula:
In other words, anything that is impermanent (as the mind is, which can be corroborated through empirical observation) is subject to suffering (again true for the mind, as can be verified through phenomenological examination) and is therefore not self.
This is an immediate flaw with the Sassatavada or eternalist mode of approaching the question of self.
On the other hand, among the Ucchedavadins or annihilationists, a stronger tendency emerged to identify self with material form. Given that the Ucchedavadins were a materialist school, their preference was to rely on evidence gathered via the bodily senses. Upon death, the body is no longer animated, and thus, according to their model, all mental and conscious life is annihilated.
Yet it is important to note that while the Buddha rejected the existence of an absolute self or ātman, he did not teach that upon death, there is the annihilation of self.
Throughout his teachings, the Buddha rejects the notion that the aggregates, either alone or together, constitute a self. Not only does he deny that any of the five aggregates, individually or in combination, constitute a self, he also extends this to ALL possible notions of an absolute self or ātman. The following scriptural quotations attest to this consistency in the Buddha’s categorical rejection of ātman as a whole:
“Sabbaṃ, bhikkhave, anattā.”
“All, bhikkhu (monk), is not self.”
Aniccādi Sutta (SN 35.34)
This is echoed by the following:
“Sabbe dhammā anattā.”
“All phenomena are not self.”
Uppādā Sutta (AN 3.137)
And further echoed by the Buddha’s strong admonition:
“Sabbesu dhammesu anattānupassī viharati.”
“Dwell realizing that in all phenomena there is no self.”
Anattānupassī Sutta (AN 7.18)
These are unequivocal, categorical statements made by the Buddha in which he rejects the possibility of ātman in every way. In stating that ALL (Sabbaṃ; Sabbe; Sabbesu) is not self, the Buddha denies the possibility of ātman not just in the aggregates, but at all levels, rejecting the very notion of ātman all together. In letting go of the notion of a soul or self, the bonds of dissatisfactory experience are shaken and no longer is the delusion of ātman present to serve as an obstacle to enlightenment. This is the most sublime peace.
 For the recurrence of the Buddha’s categorical rejection of an absolute self via the statement “sabbe dhammā anattā” or “all phenomena are not self” see:
MN 1, 4. mahāyamakavaggo, 5. cūḷasaccakasuttaṃ (MN 35)
SN 3, 1. khandhasaṃyuttaṃ, 9. theravaggo, 8. channasuttaṃ (SN 22.90)
SN 4, 10. abyākatasaṃyuttaṃ, 10. ānandasuttaṃ (SN 44.10)
AN 3, 3. tatiyapaṇṇāsakaṃ, (14) 4. yodhājīvavaggo, 4. uppādāsuttaṃ (AN 3.137)
See also the abundance of teachings by the Buddha and his foremost disciples rejecting an absolute self in the Khuddaka Nikaya:
KN Dhp, 20. maggavaggo (KN 2.20)
KN Th, 15. soḷasakanipāto, 1. aññāsikoṇḍaññattheragāthā (KN 8.246)
KN Nidd I, 4. suddhaṭṭhakasuttaniddeso
KN Nidd I, 9. māgaṇḍiyasuttaniddeso
KN Nidd I, 10. purābhedasuttaniddeso
KN Nidd I, 11. kalahavivādasuttaniddeso
KN Nidd I, 16. sāriputtasuttaniddeso
KN Nidd II, pārāyanavaggo, pārāyanavagganiddeso, 1. ajitamāṇavapucchāniddeso
KN Nidd II, pārāyanavaggo, pārāyanavagganiddeso, 2. tissametteyyamāṇavapucchāniddeso
KN Nidd II, pārāyanavaggo, pārāyanavagganiddeso, 4. mettagūmāṇavapucchāniddeso
KN Nidd II, pārāyanavaggo, pārāyanavagganiddeso, 5. dhotakamāṇavapucchāniddeso
KN Nidd II, pārāyanavaggo, pārāyanavagganiddeso, 8. hemakamāṇavapucchāniddeso
KN Nidd II, pārāyanavaggo, pārāyanavagganiddeso, 18. pārāyanānugītigāthāniddeso
KN Nidd II, khaggavisāṇasutto, paṭhamavaggo
KN Nidd II, khaggavisāṇasutto, tatiyavaggo
KN Nidd II, khaggavisāṇasutto, catutthavaggo
KN Paṭis, 1. mahāvaggo, 1. ñāṇakathā, 1. sutamayañāṇaniddeso
KN Nett, 4. paṭiniddesavāro, 1. desanāhāravibhaṅgo
KN Nett, sāsanapaṭṭhānaṃ
KN Peṭ, 2. sāsanapaṭṭhānadutiyabhūmi
KN Peṭ, 2. sāsanapaṭṭhānadutiyabhūmi, tatthimā uddānagāthā
Each of the above contains the exact same statement in rejection of an absolute self: “sabbe dhammā anattā,” meaning “all phenomena are not self,” extending beyond the five aggregates to everything that exists.