I saw this sentence in a textbook:
これは私が大好きな喫茶店です。
I’m puzzled by the が, because I’ve learned that the thing being liked is the subject of 好き:
この喫茶店が好きです。
Cure Dolly said that the particles are the one thing whose role never changes and が always marks the subject with no exception. So what is the subordinate sentence whose subject is 私?
These are two different sentences.
Here you use 大好きな which is a な adjective, hence the が after the 私, because it’s my most likable coffee house. Perhaps beloved would fit well as an adjective here.
However, if you strip this sentence from all the fluff it’s basically - 喫茶店です。
Here if you strip the sentence you’re left with 好きです。
So there’s a difference in emphasis here.
Best way is to read about な adjectives.
But @Leebo would probably explain it better than I ever will…
In the dictionary you’ll actually find this for the definition of が like in the case of この喫茶店が好き
希望・好悪・能力などの対象を示す。
Indicates the object of aspiration, like/dislike, ability, etc.
In other words, it’s not marking the subject in that situation.
And in 私が好きなもの the が is marking the subject.
I personally can’t stand Cure Dolly, though, so I can’t speak to what she says.
Well, to be completely fair, 対象 isn’t the grammatical object, but more like… the target. Which would be the (grammatical) object too in English, but… gah, having 好き be an adjective just makes things all weird.
Whatever word you prefer to translate it as, it’s the same word as what’s used to define を’s role.
動作・作用の目標・対象を表す。
I understand the meaning of each part of the sentence, but I still don’t understand why 私 is marked with the subject marker.
So you’re saying that が doesn’t always mark the subject? I understand you disagree with Cure Dolly and now I’m at a loss what model of the Japanese language I should build in my mind 
That’s what the dictionary is saying. I’m sure someone can find 5 linguists who have 5 other frameworks though.
@Leebo already answered. But: 1) I am applying bold face in the text below to the definition of my favorite grammar dictionary that is relevant to your question: が can mark the OBJECT OF A STATE. 2) Cure Dolly was not a professional Japanese teacher/scholar. The textbooks that she often complaints about offer a much solid foundation than her rumblings. To make her logic appear coherent she would translate (5）私はジャズが好きです。as “Jazz has the property of being liked by me”… instead of “I like jazz.”
That person is Yamamoto-san.
（2）隣のうちには猫が3匹いる。
There are three cats in the house next door.
（3）あ、財布が落ちている。
Look, there’s a wallet on the floor.
（4）この本は表紙がきれいですね。
This book has a beautiful cover, doesn’t it.
（5）私はジャズが好きです。
I like jazz.
（6）外交官になるには語学力が必要だ。
You need to be good at languages to be a diplomat.
（7）彼は 10 ヶ国語ができるらしい。
I heard that he can speak ten different languages.
Attaches to a noun and indicates that the noun is the subject of an action or state, as in (1), (2), (3), and (4), or that the noun is the object of a state, as in (5), (6), and (7). When used in proverbs or idioms, it sometimes attaches to words other than nouns, as in 負けるが勝ち (Sometimes you have to lose to win).
Would it be correct to say that in these two examples the role of が is reversed, so for the adjective 好き, sometimes the liker is marked with が and sometimes the thing being liked is marked with が, and both are legit?
Now I only need to figure out when to use each
Thank you! I see there are some strong sentiments about the model here, so I’m wondering how to proceed about it 
I don’t see this as a problem with Cure Dolly’s model. In this particular sentence, jazz happens to be the object in English, but it’s actually the subject in similar structures in many other languages: jazz me gusta, me plait le jazz, джаз мне нравится and so on… So having it as a subject in Japanese isn’t so surprising to me.
What I do see as a problem now is the inconsistency: that for the same adjective 好き the role of が can be reversed. And this is now at odds with my mental model, because if we take Spanish again for example, you can say “jazz me gusta”, but if you say “yo gusto al jazz”, it would mean “jazz likes me”. But apparently in Japanese the が can be put on both sides of the relation without a change of meaning? Now this breaks Cure Dolly’s model in my mind, and I’m trying to understand how to conceptualise が again.
If you want a good explanation of が I recommend Human Japanese.
He breaks it down with simple examples.
Personally, there are some formulas that just stick after awhile without understanding the reason at first.
Sometimes using the sentence structure over time will help you understand the reason further ahead. There’s a fine balance between knowing why and simply accepting things as they are at the beginning.
You can have both が’s in one sentence, as in 私がジャズが好き, so if that’s something Cure Dolly’s model can’t deal with, it needs tweaking.
Is this a legit sentence? 
Due to the way が works, it’s unlikely to be uttered on its own (it would be far more likely in a relative clause, where は is typically not allowed), but there’s nothing ungrammatical about it.
Super interesting! This led me to this answer that says that “Predicates like 好きだ are called “double-ga” or “affective” predicates”:
He brings this sentence as an example, similar to the one you brought:
僕が姉がくれた本が好きですよ。
where both が in bold are two sides of the 好き…
Also found this interesting article:
どんな動物が鼻が長い？
…
象が鼻が長い
{Noses are long} is true about elephants.
Elephants have long noses.
zou - focus.
hana ga nagai - topic.
Note: the whole predicate clause hana ga nagai is the topic, not just hana, hence why hana isn’t marked by wa.When you see a double nominative with multiple ga が, like above, the first subject will be interpreted as the focus.(Inoue, 2008)
With this we’ve established that the normal way of constructing double nominative sentences like this is wa plus ga, not ga plus ga, because ga plus ga only happens in questions (and other more complicated situations), and such questions are very unlikely to happen.
This is the most infuriating part. Because it’s much easier to just say we have a “topic and a subject,” than to say “we have two subjects, but one of them happens to be the topic, plus there’s a predicative clause or whatever mixed in.”
Nevertheless, I think having two subjects makes more sense. After all, with questions like the above, you do end up with two ga が, two subjects, and that does put a hole in the “topic plus subject” theory that doesn’t exist in the “large subject (that’s also the topic) plus small subject” theory.
Another insight from that article (I finally understand the reason why we’re taught not to use は in simple Xが好きだ sentences!):
This is the reason for a sentence like the one below to have a contrastive wa は instead of a non-contrastive topic:
- neko wa suki da
猫は好きだ
Cats are liked.
- Implicature: not-cats are not-liked. Dogs are hated.
In general, when you have just one wa は, it’s a non-contrastive topic. It’s only contrastive when you have two wa は, so one topic contrasts with the other, or a small subject contrast with another small subject.
But a sentence like the above breaks this principle, because it only has one wa は despite having the contrastive function.
That’s because it’s an elliptical sentence.
- watashi wa {neko wa suki da}
私は猫は好きだ
I like cats.
- Implicature: I not-like not-cats. I hate dogs.
Above, we have the full sentence with the large subject watashi. Now we end up with two wa は, and we can see that the small subject neko was marked with the second (contrastive) wa は all along.
I think the first GA particle would have to be a hyper GA particle in order to match the circadian rhythms of the zero GA particl which would mean that the sentence makes perfect sense.
But actually let me pull out my protractor and check before I make any firm claims
Yep, here’s an example of a double が in a relative clause from the aforementioned article:
As we’ve seen before, node ので comes after a relative clause. If that relative clause has a double nominative, you can’t use the subject-marking no の in it. You can only use ga が.
- {Tarou ga takoyaki ga suki na} no de
太郎がたこ焼きが好きなので
Because Tarou likes takoyaki.
Isn’t it just " the other が"、 like in 我が国 ?
So 私が大好きな物 is basically the same as 私の大好きな物 ?