Very difficult to chose THE most beautiful, but among the US coins in my collection this one ranks very high:
Either this 1917 French 10 Francs. It has a beautiful portrait and a really nice scene on the back with a seated liberty. Second up is this Baker Island 1 Dollar coin with very nice birds front and back.
Today's category will be "Most beautiful US Prooflike". I've selected this shield nickel, although there are a few contenders:
Though it's not technically a coin, this is the one that immediately came to mind when I saw the question. 1910 shooting medal from Bern, Switzerland.
As far as pure aesthetics go, for me it's always a tossup between the Mexican Caballito and the Japanese Meiji Dragons. Although this one also has a nice look:
Lots of nice stuff and while I think there are plenty of choices, for me this is number 1 in my collection. CALABRIA, Tarentum. Circa 315-302 BC. AR Nomos (20.5mm, 7.55 g, 10h). Warrior, holding shield and two spears, preparing to cast a third, on horseback right; ΣA below / Phalanthos, holding kantharos and trident, riding dolphin left; Ω above arm, Σ below arm; below, dolphin left. Fischer-Bossert Group 73, 881 (V345/R684); Vlasto 615; HN Italy 937. VF, toned, compact flan.