I think you need both - the banker's wife's blues. Because the first possessive is still valid/true when you apply the other one. It looks funny in writing but that's what I think is correct. If it were a two-word noun, without a possessive to begin with, you would only add it to the second word. I'm trying really hard to think of an example, so I'm going to use what's in front of me!: My JBL speaker's sound is quite good. JBL speaker has no possessive by itself, so you only apply it to the second word when adding the possessive

Hi In this case, the wife's belong to the banker and the blues belong to the wife - there for there are two possessive 's Hope this helps Jane :)