A United States dollar coin with a hole in a middle do exist and its struck in gold too. I am having a discussion with a friend yesterday about weird coin that we think never exist. Any coin that come out from our topic, we will Google it, to check it existence. Beside US dollar coin with hole, my friend also look up "Largest Circulation coin" but it end up to my post he have not read before; World largest current circulation coin. When we search for "dollar coin with hole", it bought us to Heritage Auction website with a pattern gold dollar coin, 1852 Ring Dollar in Gold, Judd-145, Thin Planchet.
Here a bit of information about it by Heritage Auction:
"An annular pattern with a wide opening that restricts the legends to the date, denomination, and country name. The lower half of the reverse displays a wreath. Struck in gold with a plain edge. Both thick and thin planchet variants are known of this pattern. The thicker variant weighs around 32 grains, and examples are believed to be original strikings. The restrikes weigh around 25 grains and are struck on thinner planchets.
The reason for experimentation with gold dollars (Judd-136 to Judd-148) was to provide a larger, more convenient diameter for the coins.
Gold dollars were small and easily lost, and in the early 1850s they represented a substantial amount of money for the average worker. Convenience lost out, however, to a more practical solution: The amount of silver was reduced in most subsidiary coinage in 1853, which returned the silver denominations to the channels of commerce rather than going to the melting pot. And the gold dollar diameter was increased in 1854, via the Type Two format."
This coin is one of U.S Mint trial gold coin pattern in 1852. They're trying to make a gold dollars with a larger diameter, so it cannot easily lost like the small one. Heritage sold a similar coin to this one in 2008 for over $27,000.