Yes, I think this is trying to say: ''the game of games'', without the second ''the''.
Does this mean something like "the game of all games," like the best possible game? Or literally a game about other games? Or is it just a translation exercise that doesn't really mean anything?
Yes 99.9% it would mean the best possible game. It can also mean a game about games in the very rare case that someone wants to say something like that though, like the use of the meta- prefix in English.
the phrase seems to have limited usefulness. It would only refer to the immortal game played in London in 1851.
"Αγώνας" can also mean "struggle" or "fight" (e.g. ο αγώνας για την ελευθερία = the fight/stuggle for freedom), but not "battle". Also, this is a lesson on sports and the world "αγώνας" in this context would mean game/match.