格外, 极, 极其: especially, extremely
【四05】程度副词 / adverbs
Tone colorsā tone 1á tone 2ǎ tone 3à tone 4a neutral
格外, 极其 and 极 are high-degree adverbs that go before the adjective: 格外冷, 极其重要, 极好.
格外 means "more than usual, more than the rest": it assumes a point of comparison (often 平时, or other people), as in 雨后的空气格外新鲜. 极其 is a neutral-to-formal "extremely", common in writing and speeches: 极其重要, 极其辛苦. 极 on its own only goes before a few short adjectives (好, 大, 小, 少, 高): 进步极大, 这些字极小, 会说法语的人极少.
None of the three combines with 很 or 非常: no 很极小. Do not mix them up with …极了 either, which goes after the adjective and belongs to spoken Chinese: 好极了, 累极了 (degree complements, see 极了).
Examples
今天格外冷，多穿一件衣服吧。
Jīntiān géwài lěng, duō chuān yí jiàn yīfu ba.It is especially cold today, put on another layer.
下雨以后，空气格外新鲜。
Xià yǔ yǐhòu, kōngqì géwài xīnxiān.After the rain the air is especially fresh.
这些字极小，我都看不清楚。
Zhèxiē zì jí xiǎo, wǒ dōu kàn bu qīngchu.These characters are tiny; I can't make them out.
在这个城市，会说法语的人极少。
Zài zhège chéngshì, huì shuō Fǎyǔ de rén jí shǎo.In this city, very few people speak French.
我们的老师极其认真，每一份作业都要看两遍。
Wǒmen de lǎoshī jíqí rènzhēn, měi yí fèn zuòyè dōu yào kàn liǎng biàn.Our teacher is extremely conscientious and reads every piece of homework twice.
这次比赛对他极其重要，所以他格外紧张。
Zhè cì bǐsài duì tā jíqí zhòngyào, suǒyǐ tā géwài jǐnzhāng.This competition matters enormously to him, so he is especially nervous.
Common mistakes
✗ 这些字很极小 → ✓ 这些字极小 / 这些字极其小 (do not stack 很 on top of 极).
✗ 重要极其 → ✓ 极其重要 / 重要极了 (极其 goes before the adjective, 极了 after it).