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I guess that only leaves Great Britain... ?
Yep.

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Why would this coin not be labeled as to country of origin?
For most European monarchies, the monarch's portrait and titles, coat of arms, or monogram was considered sufficient label enough. The attitude seems to have been, "If you didn't know where George V (for example) was king of, you weren't worthy to use his coins."

Besides, the country of origin is indicated: "BRITT OMN" (Abbreviated Latin for "All the Britons"), the realm the coin was struck for. Things only got confusing when the colonies started making their own coinage with the same titles.
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Was it considered 'legal tender' three pence in any
commonwealth country? In other words: Would this three pence
have circulated legally in Australia? Just curious.
Depends on the country. In Australia, for example, British coins were formally withdrawn sometime in the 1920's. So this particular coin would never have been "legal tender" in Australia.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis