I have heard people do this--make a noun that is modified with additional information (e.g. Aaron who didn't finish college's mom, The dog that is dirty's owner, etc.) possessive, but I don't think I've ever seen it in writing.
I might hear "the girls who live down the street's mother works three jobs," but written down the sentence would appear as: "The mother of the girls who live down the street works three jobs." This sounds fine to me, even better than the original sentence, but there are examples of sentences that become awkward from this rearrangement.
Is it considered wrong to use the possessive form on a noun that doesn't stand alone?
I wonder specifically about constructions in which the noun, before being put in the possessive form, has multiple words that come after it (What would you call this?). As shown in the examples above, this makes it so the apostrophe is attached to a word other than the target noun. In the examples I gave, these words are college, dirty, and street as opposed to Aaron, dog, and girls.