There have been, of course, many "first times" for things among the coins of the US commemorative series. In this post, I present those of a more general/administrative/legal nature, in a future post I will focus on the first appearance of specific design elements.
Note: These lists include only those coins authorized by Congress and approved by the President specifically as commemorative coins - i.e., authorized by a specific Public Law. The lists do not consider the many recent thematic coins the Mint has struck under its own authority (though a general authority granted to it by Congress). Also, the lists only consider the not-intended-for-circulation (NIFC) coins of the classic era and the non-circulating legal tender (NCLT) of the modern era; I have left off "controversial" circulating commemorative issues (e.g., 1909 Lincoln Cent, 1932 Washington quarter, etc.) at this time.With that, let's begin our look at "firsts" within the US commemorative coin series (classic and modern combined):
Denominations
First $0.01: 2009 Lincoln Bicentennial (Public Law 109-45 - 109th Congress, Law #45)

First $0.05: 2004 Westward Journey / Louisiana Purchase Bicentennial - Peace Medal design (PL 108-15)

First $0.10:
NYI
First $0.25: 1893 World's Columbian Exposition,
Isabella Quarter (PL 52nd Congress, Session II, Chapter 208)

First $0.50: 1892 World's Columbian Exposition, Columbus Half Dollar (PL 52nd Congress, Session II, Chapter 381)

First $1.00: 1900 Lafayette Memorial (PL 55th Congress, Session III, Chapter 424, 1899)

First Gold $1.00: 1903 Louisiana Purchase Exposition (PL 57th Congress, Session I, Chapter 1301, 1902 - PL 57-182)

First Quarter Eagle ($2.50): 1915 Panama-Pacific International Exposition (PL 63rd Congress, Session III, Chapter 13, 1915 - PL 63-233)

First Half Eagle ($5.00): 1986 Statue of Liberty-Ellis Island (PL 99-61)

First Eagle ($10.00): 1984 Los Angeles Olympics (PL 97-220, 1982)

First Double Eagle ($20.00):
NYI
First Quintuple Eagle ($50.00): 1915 Panama-Pacific International Exposition (PL 63rd Congress, Session III, Chapter 13, 1915 - PL 63-233)
(Note: US Laws were initially identified by a chapter number within the Statutes of a given Congress (see Isabella, Columbian and Lafayette coins above). Beginning in 1901, Public Laws began to be identified by a sequential number as well as a chapter reference within the overall Federal Statutes (see the Louisiana Purchase Exposition and Panama-Pacific International Exposition coins above). The chapter reference was later dropped in favor of just a sequential number (see the LA Olympic and Statue of Liberty- Ellis Island coins above). Mint Facilities
First Philadelphia Commemorative Coin: 1892 Columbian Exposition Half Dollar

First Denver Commemorative Coin: 1933-D Oregon Trail Half Dollar

First San Francisco Commemorative Coin: 1915-S Panama-Pacific International Exposition coins

First West Point Commemorative Coin - Prior to Becoming Official Branch Mint: 1984 Los Angeles Olympics Eagle ($10.00)
>>> Coins were first struck in September of 1983
First West Point Commemorative Coin - As Official Branch Mint: 1988 Olympic Half Eagle ($5.00)
>>> Coins were first struck in May of 1988
First Manila Commemorative Coin: 1936 Philippines Commonwealth
>>> Two One Peso coins and one 50 Centavos coin were struck
First Carson City Commemorative Coin:
NA
First Charlotte Commemorative Coin:
NA
First Dahlonega Commemorative Coin:
NA
First New Orleans Commemorative Coin:
NAMetals
First Commemorative Silver Coin (0.400 fine): Three-way tie - 1975-76 Bicentennial Quarter, Half Dollar and Dollar

First Silver Commemorative Coin - 0.900 fine: 1892 World's Columbian Exposition Half Dollar

First Silver Commemorative Coin - 0.999 fine: 2019 Apollo XI 50th Anniversary Dollar

First Gold Commemorative Coin - 0.900 fine: 1903 Louisiana Purchase Exposition Dollars (2)

First Gold Commemorative Coin - 0.999 fine:
NYI
First Rose Gold Commemorative Coin - 0.850 fine: 2018 Breast Cancer Awareness Half Eagle

First Commemorative Platinum Coin - 0.9995 fine:
NYI
First Bi-Metallic Commemorative Coin: 2000 Library of Congress (Platinum and Gold)
>>> Pt 0.9995 fine / Au 0.900 fine; 48% gold / 48% platinum / 4% alloy (likely copper)
First CuNi (Clad) Circulating Commemorative Coin: 1976 Bicentennial Quarter, Half Dollar and Dollar

First CuNi (Clad) NIFC/NCLT Commemorative Coin: 1986 Statue of Liberty Half Dollar

First CuNi (Solid) Commemorative Coin: 2004 Westward Journey Series - Peace Medal Nickel
Inscriptions
First Appearance of "In God We Trust": 1915-S Panama-Pacific International Exposition Half Dollar

First Appearance of "E Pluribus Unum": 1915-S Panama-Pacific International Exposition Quarter Eagle

First Appearance of "Liberty": 1918 Illinois Statehood Centennial Half Dollar
Miscellaneous
First Commemorative Coin to Accidentally Have Its Mint Mark Left Off: 1925 Ft. Vancouver Centennial
>>> The Vancouver coin was struck in San Francisco but does not include a mint mark.
First Coin to Have its Full Authorized Mintage Struck with No Coins Returned to the Mint to be Melted: 1918 Illinois Statehood Centennial Half Dollar
100,000 were authorized, 100,000 were struck and delivered, 0 returned to the Mint to be melted - Per US Mint reports
First Commemorative Coin to be Struck at Three US Mints: Oregon Trail Memorial Half Dollar, 1926-(P), 1926-S and 1933-D

First Commemorative Coin to be Struck at Three US Mints - Same Year: Three-way tie - 1935-P/D/S Arkansas Statehood Centennial, 1935 P/D/S Boone Bicentennial and 1935 P/D/S Texas Independence Centennial

First Commemorative Coin to be Struck at Four US Mints: 1984 Los Angeles Olympics Gold Eagle -

>>> Philadelphia, Denver, San Francisco and West Point
Notes: NA = Not Applicable; NYI = Not Yet IssuedStay tuned for Part II!
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