From reading this answer I come to understand that anatta means the lack of a core that can be conceived as self. If there is no permanent self, then who or what gets enlightened?
The main point of the answer you cite is that anatta is a characteristic of ultimate reality. Your question misses this and conflates two different paradigms or levels of understanding.
In Theravada Buddhism we recognize two levels of truth; conventional truth (sammuti-sacca) and ultimate truth (paramattha-sacca). Conventional truth relies on a spatio-temporal paradigm, in which there exist people, places, and things. It is the level on which "you" and "I" exist, and it is the level on which a "person" becomes "enlightened". None of this has anything to do with the characteristic of non-self, which works on the level of ultimate truth. Ultimate truth relies on an experiential paradigm, in which there exists only the momentary experience of seeing, hearing, smelling, tasting, feeling, or thinking.
So, the answer to "who becomes enlightened?" is "a being becomes enlightened." As to how that can be reconciled with the teaching on non-self, there is really no conflict, since non-self describes a characteristic of ultimate reality, at which level concepts like "being" or "enlightened" have no place.
The anatta concept doesn't say that there is no self - it says "not self" as in "all phenomena is not self". There is a self, but it is not permanent and is not independent from the inter-workings of the senses, sensation, perception, mind and consciousness.
For e.g. a static image that is projected on the wall - it is a result of light projected onto a wall from a projector. If you dig deeper, you can explain how a projector works, how electronics and electricity works. The image does not exist permanently and independently from the wall or projector or electricity. But this does not mean that the image does not exist. Surely it exists.
So, similarly, if a person become an Arahant while living, then this "self" or "person" surely is the one who became enlightened. But this "person" is not permanent or existing independently.
To answer your question we just need to explore the concept of Anatta.
Anatta doesn't mean that there is nothing within us, there is a mind. All sentient beings have a mind, eventhough they have different bodies or are in different realms, this is all temporary.
What Anatta rejects is the idea of a soul that keeps your current caractheristics, just like we see in some movies. According to Buddhism all you can see in the mirror, your body, is nothing but your Kamma Vipaka, the consequences of your Kamma that have ripened.
Anatta also emphatizes the idea of not identifying with the body as something that belongs to us. Now you are a human being, tomorrow you could be a bee, a deva or a hell being. So essentialy your mind is just like any other mind except for the different Kamma is "carries" and creates. This mind is what can be enlighted.
But I think that it's a difficult question to answer because it's a person's view of self which causes (or co-arises with) suffering; because that view of self should/must be abandoned on the way to enlightenment; and because that self is impermanent (even momentary) and enlightenment is opposite.
A dichotomy between self (which needs enlightening) and Bodhi (which means enlightened understanding) is expressed in the poems associated with Shenxiu and Huineng.
Only Shenxiu wrote a poem, anonymously on the wall in the middle of the night. It stated:
身是菩提樹， The body is a Bodhi tree,
心如明鏡臺。 The mind a standing mirror bright.
時時勤拂拭， At all times polish it diligently,
勿使惹塵埃。 And let no dust alight.
After having read this poem aloud to him, Hui-neng asked an officer to write another gatha on the wall for him, next to Shenxiu's, which stated:
菩提本無樹， Bodhi is fundamentally without any tree;
明鏡亦非臺。 The bright mirror is also not a stand.
本來無一物， Fundamentally there is not a single thing —
何處惹塵埃。 Where could any dust be attracted?
The Platform Sutra describes the first of these as a way to practice ("You should all recite this verse so that you will be able to see into your own natures. With this practice you will not fall into the three evil ways.") but not as having attained enlightenment.
The second one is associated with "knowing the original mind" or "knowing the mind and seeing its true nature".
Same as the giver, the act of giving and the given. All three must be present to complete what is called "give". Similarly, if there is no self, the "self" is missing and thus there is no enlightenment.
The concept of self identity or perception of self is a source of misery, as you continuously compare your perception with reality and get disappointed or carried away when there is a gap.
Since perception of self identity is a source of misery you should try to:
- reduce the polarity of the perception of self. One such way is to contemplate there is no self (at least as perceived within the framework of conventional wisdom.). This will not completely remove perception. There will be residue or worse, a perception with the opposite polarity.
- practice meditation to experience the cessation of perception
Here conventional wisdom is "wisdom" that can be perceived, comprehend, though within the field of perception.
Ultimate reality is something that you experienced with the cessation of perception, something leading to the cessation of perception, not comprehended by though or perception.
Self identity is a perception or notion that we have built in our minds. With the cessation of perception the notion or view of self dissappear. So in the realm of ultimate reality there is no self. But this is due to a paradigm shift in our thinking due to cessation of perception and notion but the process which we perceived or identified as a entity before still continues.
While you have not experienced the cessation of perception, it is wise not to ponder on the subject of self and not self too much as this itself maybe building perceptions of the opposite polarity.
This very same realization is enlightenment.
You are a continuum of phenomena that has arisen in the universe through cause and effect, in a very real way you are the universe. When one becomes enlightened one simply knows this and when you die, you return to the universe just as a flame goes out. Ask yourself, where does a flame go when it goes out? think about it for a while. Get a match and see for yourself. It simply ceases to be, but everything that enabled the flame, was inherent in the flame still exists everywhere. It just no longer takes form, as the conditions for it have been cut off. Within you, there is a cause for body, a cause for consciousness, a cause for perception, a cause for mental formation and a cause for feeling. Namely, craving. When you become enlightened you realise this and craving ends, cutting you off from arising again, just like the flame. The question who is it that gets enlightened? is an oxymoron. Instead try asking yourself the question, why do I feel like 'I am'?