Absurd, outrageous, or unthinkable; something that is far beyond expectation or common sense (can also mean terribly bad).
そんな要求はとんでもない。到底受け入れられない。
That demand is outrageous. There's no way we can accept it.
犯行の手口はとんでもないほど巧妙だった。
The method of the crime was unthinkably clever.
彼の説明はとんでもない誤解を招いた。
His explanation caused a terrible misunderstanding.
An emphatic interjection meaning “absolutely not!”, “no way!”, or “far from it!” used to deny or reject something strongly.
「君が犯人だって？」—「とんでもない！そんなことあるわけないだろう。」
"You say you're the culprit?" — "No way! That's impossible."
お礼なんて、とんでもない。助けになれてよかったよ。
Don't mention it — not at all. I'm glad I could help.
そんな高価なプレゼントを受け取るなんて、とんでもない！
There's no way I can accept such an expensive gift!
(As a polite response) "Not at all" / "Don't mention it" — used to downplay thanks or praise.
「助けてくれてありがとう」「とんでもない、いつでも言ってね。」
"Thanks for helping me." "Not at all — tell me anytime."
賞賛を受けて彼は照れて「とんでもない」と笑った。
Receiving the praise, he smiled and said modestly, "No, not at all."
「料理が美味しかったよ」「とんでもない、まだまだですよ。」
"The food was delicious." "Not at all — I've still got a lot to learn."