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Literally, 極 means 'extreme.' In Cantonese, 北極 (extreme north) and 南極 (extreme south) refer to the North and South Poles. When used as an aspect marker, 極 conveys that something has been done to its fullest extent or completely maxed out, often without achieving the desired result or the result is unexpected.
It typically follows the structure '都...' and is usually translated as 'no matter how hard/much (someone + verb)' or 'keep (verb-ing sth) but'. Its Mandarin equivalent is '怎麼...都'.
Note that 'no matter how much (someone + verb)' can be literally translated as "無論(sb-verb)幾多' or '無論(sb)點(verb)都..." (無論 - how matter). We translate 極 as 'no matter how hard/much (someone + verb)' since there is no equivalent word in English to modify the verb that can express the same meaning.
【subject - verb - 極 - (obj./n.) - 都 - result】
Unexpected Result
點解你食極都唔肥嘅？
Why do you never get fat no matter how much you eat?
(You never get fat even if you eat to an extreme extent.)佢跑極都唔攰
S/He never gets tired no matter how long s/he runs.
(S/He never gets tired even if s/he runs to an extreme extent.)隻貓食極都唔飽，唔怪得之咁肥
No matter how much the cat eats, it is never full. No wonder it is so fat.
(The cat is never full even if it eats to an extreme extent.)
Undesired Result
我搵咗佢好耐，但問極都冇人知佢去咗邊
I have been looking for him/her for so long. I keep asking but no one knows where he/she went.我搵咗嗰件衫好耐，但搵極都搵唔到
I've been searching for that shirt for so long, but no matter how hard I look, I can't find it.
(lit. ..., no matter how hard I find, I can't find it.)我諗極都諗唔到個答案
No matter how hard I think, I can’t figure out the answer.我真係冇做過，但我講極都冇人信我
I really didn’t do it, but no matter how much I explain, no one believes me.
(lit. ..., no matter how much I say, no one believes me.)我話極你，你都唔聽我講
I keep telling you, but you never listen to me.