for the life of one
(colloquial, only with a negative) Even if one's (own) life depended on it; at all.
For the life of me, I can't remember the name of that song.
Not for the life of him could he remember it.
They were unable, for the life of them, to remember.
Synonyms: for toffee, for sour apples, for beans, to save one's life