I love all of the rooting fanboys saying that you need to root but they can't give even one example of an app that you can only get from the android market. Can you guess why that is? It's because many of them have no clue. Most of the responders probably don't even own a Fire. Of the ones who do own one, most of them probably rooted immediately without ever checking out the Amazon market because rooting is what the cool kids do and they're far too cool to be stuck with the Amazon market. If they use THAT, somebody might think they're a noob.
If you're curious, just read the app reviews on Android Central or one of the other Android blogs. Whenever you're bored, go read a couple of app reviews. If you see an app that looks like you want/need it, go look for it in the Amazon market. If it's there, you win. If it's not there, then you have your first data point. Start making a list of all of the apps that you notice like that. If you find one that makes you scream "WANT!!!!", then it's time to root your fire. When you finally have a big list of apps that you would like to have but don't absolutely need, then it's time to root your Fire. The definition of "big" is totally up to you. Maybe you'll find 3 apps that are making you drool and you'll be rooted by tomorrow night. Or maybe it'll be a couple of weeks from now and you'll do it because you've filled an entire sheet of paper with games that you're dying to try.
The biggest thing is that you're going to have to base it on your opinion, not somebody else's opinion. For example, there are a couple of million people who think that "Beautiful Widgets" is amazing. When I downloaded it, I kept trying to figure out what I was missing. I don't have any problem with other people liking it, and I hope that it makes them love their droid. For me, it was distinctly meh.
So, what's the must-have app that doesn't exist on the Amazon Market? Beats me. They have Angry Birds, Fruit Ninja, and The Moron Test. Does anybody actually run anything else?
Dan
p.s. My DroidX is rooted because otherwise the damned button lights are psycho and won't turn off. My Galaxy Nexus is not because why bother? My GTab is rooted because the stock firmware is beyond sad. My Archos is not because it's mostly just a remote control for the Squeezebox and a way to do quick google searches from the couch.