Is this list complete? Fugio cent (1787) Large cent (1793–1857) - Flowing Hair cent, chain reverse (1793) - Flowing Hair cent, wreath reverse (1793) - Liberty Cap cent (1793–1797) - Draped Bust cent (1796–1808) - Classic Head cent (1808–1814) - Coronet cent (1816–1857) - Matron Head cent (1816–1839) - Braided Hair cent (1839–1857) Flying Eagle cent (1856–1858) Indian Head cent (1859–1909) Lincoln cent (1909-present) - Wheat back (1909-1958) - Memorial back (1959-2008) - Lincoln Commemorative backs (4) (2009) - Shield back (2010-date) I did not include changes in composition (e.g., 1943 Lincoln steel cent), minor changes to design (e.g., Indian Head Cent), or varieties. I realize that there will be some disagreement on what constitutes a "minor design change".
Fugio cent is in a gray area, as it comes before the establishment of the mint and ratification of the Constitution that gives Congress the authority to coin money.
If you want to include patterns: http://coinauctionshelp.com/Pattern_Guides/onecent_patterns_Home.html#.VWXja89Vgg3 Also, 1 centavo coins for the Philippines. http://coinquest.com/cgi-bin/cq/coins?main_coin=1086
There had been calls for the establishment of a mint before the Fugio (and I believe even legislation to establish one), and yes it came before the Constitution, but we did have a central government at the time under the Articles of Confederation and it did have the authority to coin money and to establish monetary standards. (It's just that under the Articles the states also had the authority to strike coins as well, but they were supposed to conform to the central government standards.) So you could include the Connecticut, Massachusetts, and New Jersey coppers as well, but not the Vermont's. Vermont wasn't part of the United States. It was an independent Republic.
Like all type sets, collectors gets to define what they consider "complete". In this case my definition would be different.
My personal "One Cent" type set is composed of business strikes struck by the US Mint for the years ranging 1793 through 1958 (the end of the Wheaties). Here's my list (I'm only missing one representative for each group): 1793 Chain Reverse 1793 Wreath Reverse 1793-96 Liberty Cap 1796-1807 Draped Bust 1808-1814 Classic Head 1816-1835 Liberty Head/Matron Head 1835-1839 Liberty Head/Matron Head Modified (Young Head) 1839-1857 Liberty Head/Braided Hair 1856-1858 Flying Eagle 1859 Indian Head/CuNi Laurel Wreath 1860-1864 Indian Head/CuNi Oak Wreath with Shield 1864-1909 Indian Head/Bronze 1909-1942/1944-1958 Lincoln Wheat Ears, Bronze 1909 Lincoln Wheat Ears/VDB, Bronze 1943 Lincoln Wheat Ears/Zinc Coated Steel
Of all the obv designs, notice the Lincoln is by far the longest running....100+ years. It illustrates the problem created by having political figures on our coins, once on, you can't get them off.