"원 없이" shares the word "원(願)" with the more familiar "원하다" (want, desire). 원 is a noun meaning "wish", "desire", and appears in some expressions independently. "원 없이" can be confusing though because it appears like "without a wish" at first. What it actually means is "so there will be no more wish/desire left", that is, "all you want without reservation". So "원 없이 먹어" is "eat all you want (holding back nothing)". Some example expressions using "원" as a noun: - 원을 들어주다: grant a person's wish. - 평생의 원을 이루다: realize one's life's wish. - 모든 일이 원대로 되었다: everything turned out as I wished.