This would be great if you wanted to talk to yourself without anyone understanding what you’re saying, for example, or if you didn’t want anybody to know what you were writing down. It might also be useful if you enjoy lots of foreign language media (see music, tv shows, movies) and want to experience the “original” rather than through dubs and subs.
Personally, I would go for either Russian or Estonian, both of which are because I frequently listen to lots of music that are in those languages, and they aren’t spoken by any of my friends.
I realize this is a bit pedantic, but if you live in a city, there aren’t really languages that aren’t spoken in your region.
But for my choice, definitely Aztec. There’s nothing more fun to say than Aztec words. Tlachtli. Quetzalcoatl. Axolotl. Tlapalpoyactic.
(That last one is the word for the color “orange”)