to carry to extremes; to go to the end of something; to reach the limits of something
to reach the peak of something (e.g. luxury, hardships, etc); to be extremely (e.g. busy)
to exhaust completely; to have nothing left (e.g. to say)
「極める」 has another reading with different meanings.
彼は技術を極めた。
He mastered the skill.
彼はその道を極めるつもりだ。
He intends to master that path.
その美しさは極めている。
Its beauty is extreme.