极其
jíqí · HSK 4
extremely
极 = extreme/utmost + 其 = (intensifier particle); together: 'to the utmost' = extremely
极 means pole/extreme (as in 北极 North Pole). 其 is a classical particle adding emphasis. The compound literally reads 'extreme-its', i.e. 'to its extreme degree' — hence 'extremely' or 'absolutely'.
极 — wood radical 木 + 及 (jí, phonetic)
极 is a semantic-phonetic compound: 木 (wood) hints at the meaning, since the original sense was the ridgepole — the highest beam of a roof. 及 supplies the sound 'jí'. From 'topmost beam' the meaning extended to 'extreme, utmost, pole', as in 北极 (North Pole).
其 — top 八 + winnowing basket 其 (originally 箕)
Originally a pictograph of a bamboo winnowing basket, now written 箕 with the bamboo radical added on top. The original form 其 was borrowed phonetically to write the pronoun and particle meaning 'his/her/its' or 'that.' This loan use became dominant, so a new character 箕 was created for the basket meaning.