The button's finder, Gberl001, has asked me to comment. I saw the button the same day it was posted, but didn't reply because Nova Treasure had already correctly identified its time-period. But because Gberl001 wants my input, here it is.
This TYPE of US Artillery button, showing an eagle standing on a cannon inside an oval-shaped rim, is shown in the button-book by Alphaeus H. Albert, who says this TYPE dates from 1808 to 1821. However, the exact version found by Gberl001 is not shown in the Albert book. His button has three cannonballs in a stack which is not wider than the cannon's wheel, the cannon-part called its "trail" has a distinctive angular bend, and the eagle's beak is wide-open. That COMBINATION of characteristics is not seen on any of this type of button in the Albert book (nor in Warren K. Tice's button-book). So, Gberl001 has found what button collectors call an "unlisted" variation.
Also:
Nearly all of this type of US Artillery button were manufactured in Britain, because prior to the mid-1820s the infant American button-making industry was not yet capable of MASS-PRODUCING "rolled" thick sheetbrass buttons with an ornately-detailed emblem. But Gberl's unlisted variation's backmark appears to show a tiny raised-wings American eagle (below the wreath), which indicates it is an American-made US Military button. As I said above, American-made rolled-BRASS ones from prior to the 1820s are comparatively quite rare. I suspect that is why the Albert book does not show Gberl's button... at the time Mr. Albert wrote (and later, updated) the book, he did not know that the exact version found by Gberl existed.
Also:
As mentioned above, this TYPE of US Artillery button dates from 1808 to 1821. But being an American-made one instead of British-made indicates it was manufactured sometime after the start of the War Of 1812 (with Britain). Of course, during the war, British-made buttons were not imported into the US, so we had to manufacture our own. Afterward, Americans' dislike of Britain due to the war caused them to prefer not to buy British-made products, for about 10 years after the war's end.
Holy moly! I expected a detailed response but this blew my mind. I too, found that this exact button was not listed but I found a different book, it is viewable on Books.Google.com called "The Emilio Collection of Military Buttons" and can be seen
here. I am no research expert so this may be completely off the trail but it listed, specifically, the front of my button including an image. The excerpt, I have transcribed here:
Luis Fenollosa Emilio
15 Army, Artillery, officer, 1794-1802; solid; flat; gilt;
22. A beading on outer edge; within an oval an eagle, head to right, on fieldpiece, 3 cannon balls in pyramid beneath. Reverse: “Extra Rich Orange.” See plate 1.
From Sackett’s Harbor, N.Y. Engraved here on imported blank. Many military relics both of the army and navy have been found about the works, barracks, gardens and old navy yard at this important post during the War of 1812 and since. Ebenezer Stevens of New York City furnished artillery buttons in 1800 which were probably of the above kind.
Apologies to the author if any mistakes were made in the transcription of this excerpt.
This, coupled with the multiple images found during my searches and the fact that none matched my backing design made me think I might have found one of these types furnished by Ebenezer Stevens but unfortunately, I could not much on Mr. Stevens, let alone anything on his manufacturing of buttons.
This was the point at which I posted online to find out if I had some "unique" version. I contacted TheCannonballGuy when I didn't get a definitive answer on whether this was some unique version.
All that being said, no disrespect was meant toward nova treasure who, as TheCannonballGuy stated, had correctly identified the time period. I simply forgot to specify that I also wanted to know if this was a unique version of this button. Also, now I am extremely excited with my find, artifacts are my major drive for metal detecting, I love finding them and wondering how rare it is. Then I get another drive boost in my research efforts because I learn so much along the way. My buddy goes for the silver and coins so it's a nice balance ha ha. This was a long post, sorry if I made any mistakes, as always, I appreciate everyone's help.