I thought of this question when I've read the one about mulled beer. I thought heating the beer would make it lose carbonation, but then I've read somewhere that you have to carbonate the mulled beer as well.
So are there any beer styles that are served without carbonation or nitrogenation? I myself think that there are probably none, but I'm no beer expert (yet :D).
Historically beer was almost definitely still (flat) for thousands of years. Before the discovery/invention of force carbonation methods, all beer was carbonated naturally via bottle or cask conditioning. But people were brewing alcoholic beverages commonly referred to as beer in antiquity, and evidence from these cultures (ancient China, Neolithic culture, etc) suggests that they were doing this in big stone and earthenware pots and jugs, which may have had no lids or loose lids. Their vessels probably could not have withstood the pressure of carbonation even if they were using wax or cork to seal the vessels.
So if you were to brew a historical style and you wanted to be truly accurate, then you would not carbonate it. But the modern historical interpretations usually are carbonated (like Dogfish Head's ancient ales), because they are selling them and Westerners generally prefer them :)
There is one "modern style" (e.g. a style you'd find in the BJCP style guide), that can be served with no carbonation, namely straight (unblended) lambic. Several have little or low carbonation, (e.g. barleywine) mentioned in other answers. Sahti also has very little carbonation.
But people do still drink flat/still beer all over the world, for example there's some uncarbonated corn beers that are popular, like chicha in Latin America (Dogfish Head made a Chicha inspired beer) and umqombothi in South Africa. There are other names for similar beers in other regions. These are typically homebrewed.
What you're most likely to find everyday are not beers that are flat, exactly, but are very low in carbonation. Barleywine ale is one of these styles, and though it will typically have some carbonation, it will be very little. In the UK, real (or cask) ale is another style with very little carbonation.
The only style I can think of off the top of my head that is really and truly uncarbonated are the super high alcohol beers like Boston Beer's Samuel Adams Utopias and some of the various high-alcohol freeze-distilled* varieties. To me, these styles have nearly as much in common with fortified wine or spirits as they do with beer, and they're not easy to find, so they're a bit of an outlier.
*Distillation requires heat, so the process of freezing to concentrate alcohol in a beer isn't really technically distillation, but hey, that's what it's called, so when it Rome...
Cask beers have very low levels of carbonation, enough that one could almost consider them to be flat. This is due to the fact that they aren't served under any pressure.
I home brew ales and stouts all the time and have never carbonated or even bottled it. I have never missed the bubbles or the constant burping. In fact, I am enjoying a glass of English Bitter right now. Tasty and refreshing. Five gallons of beer so that I can dip a jug whenever I want, and no messing with bottles. By the time you finish the 2nd one, you will not miss the carbonation.
