It bears repeating that our self is not exhausted by the five aggregates which means we have the ability to transcend them. Still we continue to grasp the aggregates, cling to them and imagine ourselves to consist in them even though the aggregates are suffering (S.iii.158)
(edit) S.iii. 158 Quite clearly the focus in the teaching is the craving for, and clinging to, the aggregates as the source of suffering, and not the aggregates themselves.
103 (1) Portions
At Savatthi. "Bhikkhus, there are these four portions. What
four?  The portion of identity, the portion of the origin of
identity, the portion of the cessation of identity, the portion of the
way leading to the cessation of identity.
"And what, bhikkhus, is the portion of identity? It should be
said: the five aggregates subject to clinging. What five? The form
aggregate subject to clinging, the feeling aggregate subject to
clinging, the perception aggregate subject to clinging, the volitional
formations aggregate subject to clinging, the consciousness
aggregate subject to clinging. This is called the portion of identity.
"And what, bhikkhus, is the portion of the origin of identity? It
is this craving that leads to renewed existence, accompanied by
delight and lust, seeking delight here and there; that is, craving
for sensual pleasures, craving for existence, craving for extermination.
This is called the portion of the origin of identity.
"And what, bhikkhus, is the portion of the cessation of identity?
It is the remainderless fading away and cessation of that same
craving, the giving up and relinquishing of it, freedom from it, nonreliance
on it. This is called the portion of the cessation of identity.
"And what, bhikkhus, is the portion of the way leading to the
cessation of identity? It is this Noble Eightfold Path; that is, right
view ... right concentration. This is called the portion of the way
leading to the cessation of identity.
"These, bhikkhus, are the four portions."
... which is the first noble or ariyan truth. There is nothing redeeming about these aggregates. They equated with Mara the demon.
"When there is form, Radha, there might be Mara, or the killer, or the one who is killed. Therefore, Radha, see form as Mara, see it as the killer, see it as the one who is killed. See it as a disease, as a tumor, as a dart, as misery, as really misery. Those who see it thus see rightly. When there if feeling ... When there is perception ... When there are volitional formations ... When there is consciousness, Radha, there might be Mara, or the killer, or the one who is killed" (S.iii.189) (trans. Bhikkhu Bodhi.)
When we cling to the aggregates not only are we clinging to suffering but we are clinging to Mara. Some have yet to realize that it is easier to know what I am not (e.g., the five aggregtes) than what I am. With the former we can thoroughly abandon desire for whatever is not our self whereas to imagine what the self or my self consists in, is never certain. More than likely it is one of the aggregates which ends up being a wrong view of my self. What I imagined my self to consist in turned out to be transient and suffering rather than permanent.
It is difficult to see where the current theory is going that, fundamentally, there is no self
. It paints a very unsystematic and confusing picture of the Buddha's teachings which sounds more like materialism. It is much easier to take the position that my self is greater than the sum of the aggregates; which is not bound by their limitations. This ends up making Buddhism systematic and not confusing.