到底
とうてい · toutei · JLPT N1 · adverb
(cannot) possibly
到 = reach + 底 = bottom: can't reach the bottom, so (cannot) possibly
到底 literally means reach the bottom. Used with a negative, it says something cannot be done even at the deepest or furthest limit, so 'cannot possibly' fits.
到 — 刂 knife + 至 (phonetic, 'arrive')
到 is a semantic-phonetic compound. 至 ('to arrive') gives both meaning and sound, while 刂 (knife) was added later, likely as a phonetic reinforcer or to distinguish the written form. The core sense 'to arrive, reach' comes directly from 至, and 到 has carried that meaning since early script forms.
底 — 广 (shelter radical) + 氐 (phonetic 'dǐ')
底 is a semantic-phonetic compound. The 广 radical depicts a roofed structure or shelter, suggesting the bottom or base of a building. The component 氐 supplies the sound 'dǐ' and originally meant 'low' or 'base,' reinforcing the meaning. Together they convey 'bottom,' 'foundation,' or 'underneath.'