Here is an example of the absolute rarest aviation badge in the US aviation field. There were only 27 of these awarded, and according to Duncan Campbell, there may have been 28 - but that is not clear. The badge was created by General Order 39 on May 23, 1913. Originally, it was to be struck in silver to correspond with the rifle and pistol qualification badges of the era. However, it was decided that they were to be struck in 14 kt gold.
The badges were die struck at Rock Island Arsenal, and by the firm of Blackington. There are noticeable differences in the two when compared side by side. After striking, they were hand chased to bring out and finish the fine details.
An interesting side bar concerning the original die - many years ago, it was decided to strike a very limited number from the original dies, but in brass. Unfortunately since brass is much harder that gold, the die was damaged during re-striking.
I do not own this badge. The one shown was sold in the recent Duncan Campbell auction in San Franciso a few months ago.