The phrase "be gaga over" is an informal expression which means being obsessed with something or someone. There are several synonyms for this phrase, such as "crazy about," "infatuated with," "smitten with," "head over heels," "madly in love with," "wild about," and "enamoured of." These synonyms convey the same feeling of intense passion and adoration towards someone or something that "be gaga over" does. These words can be used interchangeably depending on the context or the tone of the sentence. Therefore, if you want to express your love or admiration for someone or something, these synonyms can come in handy to give a broader variety of words to choose from.