の至り・の極み・極まりない・極まる: the difference
の至りN1
Meaning
Utmost, Height of, Extreme, Heartiest, Such a, Very, really, Completely, Deeply
Past papers
Example
演説：「特別賞を受賞することになり、光栄の至りです。」
Speech: 'It is the greatest honor to receive this special award.'
の極みN1
Meaning
Ultimate, Extreme, Utter, Epitome, Peak
Past papers
Example
タイトル：「シンプルの極み！絶品『ハンバーグ』の作り方」
Title: 'The epitome of simplicity! The recipe for a perfect Hamburg steak.'
Meaning
Extreme, Extremely
Past papers
Tested 3 times in past papers
Example
犯罪を犯すのは愚劣極まりないことだ。
It is extremely foolish to commit a crime.
How to tell them apart
「極まりない／極まる」: after an adjective stem; mostly a harsh judgement on the nature of something: 「失礼極まりない」, 「危険極まる」.
「の至り」: after certain feeling words; the deepest inner feeling from a favor received or a mistake: 「光栄の至り」, 「若気の至り」.
「の極み」: after a noun; a state or feeling at a peak that cannot be higher: 「贅沢の極み」, 「痛恨の極み」.