A formal, emphatic adverb used to stress the degree of an adjective or verb. Stronger than とても and closer to 'extremely' or 'exceedingly' in tone.
This issue is extremely important.
彼の説明は極めて分かりやすかった。
His explanation was exceedingly easy to understand.
This accident points to an extremely important issue.
The site is in an extremely dangerous situation.
極めて is a stronger, more formal intensifier used in writing, whereas ごく is milder and pairs with a narrower set of adjectives and nouns.
極めて and 至って are both literary-sounding intensifiers, but 極めて is more common in formal writing while 至って often adds a nuance of 'surprisingly so'.