Use ～極まりない in formal writing and set speeches to say something is extreme to the point of intolerable — almost always a negative evaluation (danger, rudeness, inefficiency, unfairness). It is the written-register counterpart of とても〜だ, common in reports, complaint letters, and公式コメント; in casual conversation it sounds theatrical. Note the counterintuitive form: despite the negative ～ない, the meaning is emphatically affirmative, and ～極まる means exactly the same thing.