"Not for the life of me" is a phrase commonly used to convey the idea of an impossibility or a situation that is completely out of one's control. Some synonyms for this expression include "under no circumstances," "no way," "absolutely not," "impossible," and "out of the question." Other phrases that can be used to replace "not for the life of me" include "I wouldn't dream of it," "it's off the table," "it's a non-starter," and "it's not up for discussion." These phrases can be used to express strong conviction and a firm resolve to not engage in a particular activity or accept a specific offer.