Your question blows my mind. Just the thought of narrowing down all the great coins out there, be it rarity or artwork, to three is bewildering. I think I would want the 1873 no arrows CC dime, a large cent flying eagle pattern(not sure on the date), and a Gobrecht dollar. That would be an amazing collection...
True it's hard to narrow down others on my short list are the 1870-s half dime and $3 the pacquet reverse $20 a 1795 no stars quarter eagle a half disme and the csa half dollar is tough to pick just 3
I would get the magical 1987 mercury dime! It's magic will be able to get me any coin I want! But seriously, 1943 Copper Cent 1856 Flying Eagle Cent Any 20-Cent Piece
I'll go with the 1909-s vdb cent, the 1916-d mercury dime , and the 1902-s morgan dollar. I'm trying to complete my 20th century US circulating coin set w/o varieties.