"My older brother likes purple the most."
In a conversation, Yes. In written Chinese, I have not seen someone write the 的 afterwards. It is acceptable in conversational Chinese because these 的、了 are sometimes just for expression of feelings or casuality, without a concrete meaning.
Note that you can also say 我的哥哥最喜欢紫色的 in conversation, in case you are trying to move the 的 around.