〜の極み
the height of, the ultimate in
Noun + の極み(だ)
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の極み means "the peak, the ultimate": 贅沢の極みだ, "it's the height of luxury".
It follows a noun and describes a state at its extreme, positive or negative: 贅沢, 疲労 ("exhaustion"), 悲しみ, 無責任, 光栄. It can end a sentence (〜の極みだ), or follow a verb like 達する ("reach").
It's more descriptive and flexible than の至り, which is limited to a few humble set phrases.
Compare 極まりない, which follows な-adjectives and is almost always critical. 極み is also used in marketing: 美の極み, "the ultimate in beauty".
Examples
一流ホテルに一か月泊まるなんて、贅沢の極みだ。
A month in a top hotel is the height of luxury.
三日間寝ずに働いて、疲労の極みに達していた。
After three days of work without sleep, I'd reached the limit of exhaustion.
友を失い、彼は悲しみの極みにあった。
Having lost his friend, he was in the depths of grief.
こんな大きな賞をいただけるとは、光栄の極みです。
Receiving such a big award is the greatest honour.
約束を破って謝りもしないとは、無責任の極みだ。
Breaking a promise and not even apologising is the height of irresponsibility.