I've looked other places online to try and find the answer to this question and was unable to confirm if I should use an apostrophe for both words or not. It seems awkward to do so but grammar rules appear to tell me otherwise. The situation I'm thinking of is one like this: "Mary's sister's lasagna" or "Joe's dog's bone." From the rules I've read I'm pretty sure that both words would need an apostrophe because they are both possessive, and each in their own right possessing individual things: it's Mary's sister, and the sister's lasagna. If it were Mary and her sister's lasagna, the apostrophe could go after 'sister' because they both are sharing possession of the same thing.
Grammatically, you can string possessives along as far as you want. Stylistically, though, you might consider "the lasagna made by Mary's sister" and "the bone that Joe's dog is chewing on" (or "the bone in the front leg of Joe's dog").
Dark Helmet: Before you die, there is something you should know about us, Lone Starr.
Lone Starr: What?
Dark Helmet: I am your father's brother's nephew's cousin's former roommate.
Lone Starr: What's that make us?
Dark Helmet: Absolutely nothing! Which is what you are about to become.
Correct. It does seem awkward, but your grammar is unassailably correct.