In a comment on this answer, Cort Ammon states
I have heard interesting solutions to these issues involving encrypting something with a proper tested algorithm first (not second), and then encrypting it with your home brew solution with an unrelated key. It seems legit, but I can't actually claim I've analyzed that particular configuration to see if it actually works or not.
Why does order matter? What makes encrypting with AES and then re-encrypting with the Caesar Cipher better than encrypting with the Caesar Cipher and then re-encrypting with AES?
