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Used to work with a Fijian chap, built like brick sh*t houses!
Some of the US National Guard regiments claim histories that are longer than the 3rd Regiments (Old Guard), but they are not really active. Wikipedia claims the oldest is the 182nd Infantry Regiment (182nd Infantry Regiment (United States) - Wikipedia, the free encyclopedia). Wikipedia also claims the [Connecticut] Governor's Foot Guard traces back to 1771 (Governor's Foot Guard - Wikipedia, the free encyclopedia). I think that calling the latter a "military" unit is now very shaky, as I do think it no longer undergoes any military training.