The consensus appears to be the Arkansas in Colorado. I was there this summer and used an outfitter out of Buena Vista. There are more options in Buena Vista than in Canon City. In BV, there are three stretches of river that range from class II to Class III-IV. The stretch of river called The Numbers is class III-IV+ and can get hairy. The Royal Gorge trip out of Canon City is class III-IV, too, but probably shades a bit towards class III because, though the water is big, there's little technical skill necessary to navigate the rapids.
Rapids classification is based both on the size of the rapid and the technical skill needed to safely navigate the river. There's much, much, much bigger water somewhere like on the Kennebec River in Maine during big seasonal releases but the rapids are only class III because they're pretty straightforward. The Colorado through the Grand Canyon, too.
A commercial outfitter will rarely (never?) advertise and a stretch of river as class V and won't take commercial trips down water that big. They can't get insurance for class V commercial operations. The risk of loss is too great for most insurers of commercial rafting companies.
You'll get gorgeous views on the Royal Gorge trip and more thrills on The Numbers. I guarantee you that someone in your party, if you're a novice rafter, will swim The Numbers and it won't be pretty. If you're the one in the boat, it's hilarious. If you're the poor guy swimming, it's awful.
And just to make sure, get a guide and go through a legitimate outfitter. Do not attempt to raft alone. Bad things will happen. I've seen more than one private party trip with novice rafters have to be airlifted from the side of the river. The cost of an outfitter and gratuity is much, much less than a helicopter extraction.
Edit: copy & paste typos
