The following is excerpted from Clifford Mishler’s Coins: Questions & Answers
Q: What was the purpose of the $3 gold piece?
A: Sometimes referred to as “the rich man’s three-cent piece,” it is widely assumed that the $3 gold piece was issued to facilitate the purchase of three-cent stamps by the hundred. This belief is supported by the fact that the silver three-cent piece was issued to make the purchase of the stamp less bothersome for the public, and by the discontinuance of the $3 gold in 1889, the same year the nickel three-cent piece was discontinued, but this is pure conjecture. Never minted in substantial quantities—approximately 62 percent of the total production dates from the first six years, 1854 to 1859, with another 23 percent being inexplicably produced in the years 1874 and 1878—as the coin never enjoyed extensive circulation, it is readily available today in high grade circulated and mint state conditions.
One of the great regular issue coinage rarities of the United States is the 1870-S $3 gold piece. Only one example of that coin is known to exist, part of the collection formed by Harry W. Bass Jr., which is on display at the Edward C. Rochette Money Museum at the American Numismatic Association headquarters in Colorado Springs, Colorado, although another which has never been located is rumored to have been placed in the cornerstone of the old San Francisco Mint. Interestingly, while the San Francisco Mint produced modest quantities of $3 gold pieces from 1855 to 1857 and again in 1860, the only subsequent year of record for the coin at the West Coast mint was 1870.