What is the most distinctive Company/Battery/Troop sized unit in the US Army?

Units of the MA Guard have existed since 1636 and have participated in almost every conflict that this country has been in. The 772nd MP Co was originally organized in 1638 as the Cohannet Train Band, Plymouth Colony Militia, and is the is the oldest company-sized unit in the Army National Guard. It's also one of only nineteen Army National Guard units with campaign credit for the War of 1812.

When it was organized as Company C, 4th Regiment, it was called into federal service twice during the Civil War; for three months in 1861, fighting in the Battle of Big Bethel, Va.; and for nine months from Sept. 1862 to August 1863, taking part in the siege of Port Hudson, La.

after the war as Company F, 3rd Regiment, the unit was converted to coastal artillery in 1897. During World War I, the Taunton unit served as the 24th Company, Coast Defenses of Boston. In 1920 the unit converted to field artillery as Battery F, 101st Field Artillery.

The battery was ordered into federal service in January 1941 when the 26th Infantry Division began training prior to World War II at Camp Edwards. Redesignated again as Battery C, 212th Armored Field Artillery Battalion, the Taunton unit, as part of the 6th Armored Division, fought in Europe. It took part in campaigns in France, Belgium and Germany as part of General Patton's Third Army.

In December 1967 the unit received its current designation as the 772nd Military Police Company. It also consolidated with the Attleboro unit which as, Company I, 101st Infantry, was awarded the Presidential Unit Citation for gallantry in action at Lorraine, France.

On 7 January 1991 the 772nd was ordered into federal service for duty in the Gulf War. It provided reconnaissance and security for main supply routes.

In 2002 they deployed in support of operation enduring Freedom in Bagram and Kabul Afghanistan providing base security. In 2009 the they deployed to Al Kut, Iraq, where they spent the year training the local police forces and assisted in Military operations.

America's History is more than the Civil War. My favorite would be 2nd Bn, 308th Infantry, the "Lost Battalion"

Really good question. One thing to keep in mind is that the definition of a troop sized element has changed over time. For, example, while there were companies within civil war regiments, the regiments often functioned and were identified more like our companies sized elements are now. Mostly because a Civil War era infantry company had about 50 soldiers (including officers) which is much closer in size to our definition of a platoon sized element. However a regiment had around 10 companies so one cannot say "since 19th cent company = 21st cent platoon, therefore 19th cent regiment = 21st cent company." There just isn't a common civil war era unit size that is comparable to our current definition of "unit."

Even though it's not a "company/battery/troop," my answer would be the 20th Maine. It's the closest thing I can come up with that fits the definition of a "unit" given it's historical context.