At the very beginning of the game, Vivi can participate in a Rope Jumping game in the Alexandria town square. Depending on how many jumps you manage to complete, you'll get a progressively better item. Here are the rewards:

20 jumps: 10 gil
50 jumps: Cactrot Card
100 jumps: Genji Card
200 jumps: Alexander Card
300 jumps: Cat Paw Racket
1000 jumps: King of Skipping Title