Okay... what about Gildor Inglorion, then?
Who am I and what is my name?
A chap had the same,
long long ago. –There was a Gildor in the First Age, one of Barahir's men.
Who am I and what is my name?
I played the great game,
and did I win? No! –Not sure what this would mean. Lost his friendly verbal fencing-match with Frodo and ended up giving him a straight answer? Played no real role in winning the war?
Who am I and what is my name?
Despite my great fame,
The first you can't know. –Literally, as his origin remains a mystery.