The answer to the crossword question ‘You know better than ___‘ is THAT? is ME. The phrase ‘you know better than me‘ is commonly used to acknowledge that someone else has more knowledge, experience, or authority on a particular subject than oneself. The word “that” at the end of the question is a clue to the fact that the answer is a pronoun. Therefore, when completing the phrase, the missing word is “me”, which is a pronoun that refers to oneself. Thus, the complete phrase would read ‘you know better than me‘.