〜極まる・〜極まりない
extremely, utterly (negative)
な-adjective stem + 極まる / 極まりない
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極まりない and 極まる mean "extremely, to the utmost", and they usually intensify a negative な-adjective: 彼の態度は失礼極まりない, "his attitude is incredibly rude".
Both forms mean the same thing, even though one looks negative. 極まる means "to reach the limit", and 極まりない means "without limit". 極まりない is more common.
Typical adjectives are 失礼, 危険, 不愉快, 退屈, 無責任, 不便, 残念 and 迷惑. Attach it directly to the stem, without な: 危険極まりない運転.
It's formal, strong and critical, typical of complaints and editorials. Compare の極み, "the height of", which follows nouns and can be positive.
Examples
彼の態度は失礼極まりない。
His attitude is incredibly rude.
あれは危険極まりない運転だ。
That's extremely dangerous driving.
それは不愉快極まる発言だった。
That was a thoroughly offensive remark.
退屈極まりない授業だった。
It was an unbearably boring class.
彼の行為は無責任極まりない。
What he did was utterly irresponsible.