I know that there are no official Adventure Time RPGs. Are there any fan-made ones out there? I'd be willing to accept adaptations of the characters/setting to other systems, as well as standalone games.
This has already been answered, but I'll contribute anyways with the little knowledge I have.
Savage Worlds is the system I use. It's easier to learn and run than 4e and ironically has more options. You only need the Core Book (Deluxe Edition recommended), a little imagination, and understanding of the rules. Finn the Human is a pretty straightforward character, just grab the Warrior Template and make him a child. Jake the Dog? There are stats for dogs on the Bestiary, just create a power to make him stretch and change shapes!!!
The system is very flexible and allows you as a GM to make your own content in minutes without unbalancing the game; though you might find this intimidating, believe me it's not. You'd be surprised how many things you can run with this system. There's even a Ben 10 adaptation out there to give you an idea, and if you're gonna play it with kids, it's fun and simple enough to keep them interested :)