The Logitech PowerShell Controller + Battery ($99.99), for the iPhone 5, 5S, and fifth-gen iPod Touch, is a battery pack case that doubles as a game controller with its own set of physical buttons. Read the full review.
The PowerShell Controller connects via Lightning; you just snap your iPhone or iPod in to connect. All the buttons are pressure-sensitive and analog, and there's also a dedicated pause button for games.
iOS 7 supports game controllers, and this is one of the first to take advantage of the MFi controller API. Hundreds of games are already theoretically compatible, but finding them can be a challenge. (Some good starting ones: Lego Lord of the Rings, Oceanhorn, Pac-Man, and Galaxy on Fire 2.)