extreme, utmost, furthest, final
极 centers on reaching the farthest point or limit: the extreme, the utmost, the furthest, or the final stage. All given meanings are natural applications of this idea of an endpoint.
极 combines 木 (tree, wood) with 及, which likely contributes the sound and the idea of reaching or extending. The compound suggests a wooden beam or pole reaching to an extreme point, though the exact historical development is uncertain.
A wooden beam (木) reaching (及) as far as possible: the extreme end of a pole. Picture a long wooden pole extending to the furthest point.
For キョウ, imagine a wooden beam reaching the extreme edge of a kyō (today, 今日) sunset: kyō -> キョウ, and the pole stretches to the utmost limit.