What is the rarest United States coin?

There are several coins that can arguably be called the rarest :

1913 Liberty Head nickel -- 5 known

1894-S Barber Dime -- 22 known

1853-O Seated Liberty Half (No Arrows / No Rays) -- 3 known

1866 Seated Liberty Half (No "IN GOD WE TRUST") -- 1 known

1804 Bust Dollar -- 15 known

1851-O Seated Liberty Dollar -- 1 known

1870-S Seated Liberty Dollar -- 11 known

1870-S $3 Gold Piece -- 1 known

1933 $20 Gold Double Eagle -- 1 legally owned / 10 recently seized by the Mint

I'm sure there are other coins that could be added to this list -- I haven't even touched on Proof issues, varieties, or error coins.

Actually, the " 1894-S barber dime" is wrong , it is the 1895-S barber dime

A very common question with no simple answer except there is none. Not one single coin is identified as the rarest or most valuable. Some US coins have sold for millions but are considered to be among the rarest of all US coins such as the 1913 Liberty Head Nickel (5 known) one of these nickels sold in March of 2008 for $1,840,000.00

The only legal 1933 Saint-Gaudens $20 Gold coin (13? known) sold in July of 2002 for $7,590,020.00.Then you have coins that will never be sold because they're part of the National Collection like the 1849 Liberty Head $20.00 gold piece with only 2 known coins, what would the price of one be? And the last but not least, the coins that have never be seen. The 1873s no arrows at date Seated Liberty Half Dollar 5,000 were struck and the 1873s Seated Liberty Dollar 700 were struck. Both coins are UNKNOWN in any collection. In my opinion that's rare.