An adverb used to earnestly or sincerely ask or urge someone to do (or avoid) something; akin to 'please, be sure to' or 'do take care'.
くれぐれもお大事にしてくださいね。
Please take very good care of yourself.
出張中はくれぐれも気をつけてください。
Please be very careful while you're on the business trip.
くれぐれもよろしくお伝えください。
Please give them my regards (do be sure to pass this on).
Used to indicate repetition, meaning 'again and again' or 'repeatedly'; this nuance is somewhat literary or old-fashioned.
くれぐれも何度も確認するように言われた。
I was told repeatedly to check it carefully.
彼はくれぐれも失敗しないように準備を重ねた。
He prepared repeatedly so as not to fail.