This game is about listening, but you need to represent it visually with a logo.
It's also about countries and people, but not about any one place. So if you use something cool from one country, try to blend it with something from another.
THE CORE IDEA
The world is an interesting and vast place.
The logo should look good in black and white, but can have colour. It should be understated, rather than using typical motifs: no flags, no maps.
I've attached a map of the different writing systems used in the world, and a little calligraphy. They might give you ideas about interesting shapes.
Since the game is about sounds, I'd also be open to interesting use of phonetic letters from the International Phonetic Alphabet, as attached. For example, "The Great Language Game" is pronounced: /ðə ɡreɪt ˈlæŋɡwɪdʒ ɡeɪm/
These were just quick ideas of mine. For the right idea, anything goes.
Have some fun with it. Happy designing! :)