Which of these sentences is correct: "The clock under the curtains' hour hand broke off", or "The clock under the curtains's hour hand broke off"? The actual thing being made possessive, "The clock under the curtains," is singular, suggesting that you should add 's to the entire phrase to make it possessive. This would make the latter sentence correct, but it looks funny to me. Of course you don't add 's to a plural noun already ending in "s" in order to make it possessive, but what do you do if the possessive noun phrase itself is singular but it ends in a plural noun?
I'm going to quote a comment that I think efficiently lays out some of the presuppositions that this question is based on, in order to express some disagreement with those presuppositions:
in principle one of these two options should be grammatically correct (albeit awkward), right? Which one?
It is not acceptable to add the -'s genitive (or "Saxon genitive") to all noun phrases, and in the contexts where it is acceptable, there is not always only one "correct" form. So I don't think it's right to assume that this is a binary question.
The paper The English “Group Genitive” is a Special Clitic", by Stephen R. Anderson, gives some examples of (single word) noun phrases where it is fairly clearly not acceptable to add -'s or -'.
(20) a. *These’s illustrations are more competently drawn than those’s.
b. *Of the books I lent you, two’s/some’s/many’s covers were soiled when you brought them back.
I think these examples establish that it is not impossible in principle for there to be no acceptable way of forming a -'s genitive from a particular noun phrase.
To address the specific noun phrase given in the question, I think that most speakers who tolerate the use of the -'s genitive with the noun phrase "The clock under the curtains" would pronounce the genitive construction no differently from the original noun phrase, which would support the spelling "The clock under the curtains' hour hand." Anderson brings up the topic of noun phrases ending in a word suffixed with /z/ that is not the head of the noun phrase: he says that Zwicky (1987) describes these as not taking an additional [z] sound, but Carstairs (1987) "claims that the sentences with two /z/s are often acceptable".
I would agree with tchrist's advice to "write what you say", if you use this construction at all (my preference would be to avoid using it). But overall, this is a rare construction, linguists don't give uniform descriptions of the usage, and it's simple enough to rephrase in formal contexts, so I don't see any point to being dogmatically prescriptive about there being a single "correct" usage in this context.
The rule is that you write what you say. This is a purely phonologic law.
And you say:
The clock under the curtains’ hour hand broke off.
Therefore, that is what you write. Nobody ever ever says curtainses, so you mustn’t ever write something that says that (like curtains’s). It’s curtain’s for one and curtains’ for more than one.
In writing you can convey the difference between one curtain and several, but not in speaking. If there were just one curtain you would write:
The clock under the curtain’s hour hand broke off.
This is an awkward way to say this. "The clock's hour hand" should go together because it's a logical unit. When you put "below the curtains" in the middle, a reader could easily misread "the curtains' hour hand", which is risible.
It would be better to rephrase this to:
Under the curtains, the clock's hour hand broke off.
(Not sure if you actually meant "below" instead of "under".)
The clock under the curtains has its hour hand broken off.
The clock which is under the curtains has a broken hour hand.
or any such combination.
Grammatical correctness isn't everything. Writing, first and foremost, is communication. You should do it in such a way to minimise miscommunication.
The curtain does not possess the clock. Any sentence with an apostrophe indicating possession by the curtain is wrong. Rewriting the sentence as recommended above is correct. However it is possible to have multiple possessives such as Aunt Betty's brother's car or if multiple brothers then Aunt Betty's brothers' cars.
More specific to the POp's question: you follow the same rules as other possessive.
The eye of the tiger's gaze. one eye, one tiger, one gaze
The eyes of the tiger's gaze. multiple eyes, one tiger, one gaze
The eyes of the tigers' gaze multiple eyes, multiple tigers, one gaze--as if a pack singly staring down a prey
The eyes of the tigers' gazes. multiple eyes, multiple tigers, multiple gazes
The eye of the tigers' gaze. single eye, multiple tigers, singular gaze as if the pack was staring as one
The eye of the tigers' gazes. multiple tigers, multiple gazes, but with a singular purpose