It's not a normal expression in English, either. It's a film title about a man with a split personality. In France it's translated as "Fous d'Irène", but in Québec it's indeed "Moi, moi-même et Irène".

I suppose it does lose the pun of "Me, Myself and I[rene]", though.

"Me, myself and Irene" is a stylistic idiosyncrasy of English. It wouldn't translate well and I've never heard it used in French. You'd simply say moi et Irene.

Me, myself and Irene        --  Moi, moi-même et Irene.

@Jim: are you sure that would isn't a literal translation which doesn't work in French? Similar to "everyone and their uncle" in English means "a lot of people" but chaqu'un et son oncle wouldn't work in French.