末期
まっき · makki · JLPT N1 · noun
deathbed
末 — tree 木 with extended top stroke marking the tip
末 is an indicative character: it shows the tree 木 with an emphasized horizontal line near the top to point at the treetop, the 'tip' or 'end' of the branches. From this concrete sense it extended to mean 'end,' 'final,' or 'minor/trivial' (as opposed to the root 本).
期 — 其 (phonetic) + 月 (moon/time)
期 is a phono-semantic compound. The right side 月 (moon) signals meaning, since lunar cycles marked fixed time periods in ancient China. The left side 其 supplies the sound. Together it points to a set time, period, or appointed date — meanings still carried today in words like 期间 and 日期.