Are there any situations where untempered chocolate is better than tempered?

It sounds as if you may not understand exactly what "tempering" is. You can't buy "tempered" chocolate. Tempering is the processes that makes it workable.

Chocolate is made up of a whole bunch of different fats, each with its own melting temperature. "Tempering" is, simply, raising the temperature so that they have all melted, bringing it back down to the point where some of the fats start to crystalize again, and, finally raising the temperature, carefully, to the point at which only the one "good" crystal forms. Great description here

Clearly, this is utterly unnecessary for chocolate chip cookies, fudge, ganache, etc. If you are going to make chocolate cigarettes, leaves, or nests, though, it is essential.

Tempering is a process used to give solid chocolate a uniform appearance and texture, as well as to insure that those qualities are shelf-stable. Most commercially available chocolate is already tempered to some degree so that it has a pleasing appearance and texture for customers to enjoy right out of the box/bag.

Tempering is required when the final product will be made of solid chocolate or the chocolate is a component in an unadulterated state, especially in candies that will sit at room temperature. These would include molded and enrobed chocolates such as chocolate bars, bonbons, leafs, and nests.

Recipes that add chocolate to other ingredients such as dairy, flour, and sugar can be made with either tempered or untempered chocolate because the structure of the chocolate fat crystals is no longer the dominant source of structure and texture. Examples of this include cookies, brownies, ganache, ice cream, and cakes.

There is no normal application where untempered chocolate is required and tempered chocolate could not be used. The only application I have been able to think of where untempered chocolate would be desired/required is something made for the express purpose of training and/or testing chocolatiers or chocolate tasting professionals.

No, there aren't. Nobody likes untempered chocolate.

Try taking a bar of pure chocolate. It's tempered. Now heat it up to 50-60 Celsius. Let it cool down gently.

What you have now is distempered chocolate. There is no accounting for taste, of course, but whenever that happens, people tend to cry out "Help! My chocolate is ruined!". I have never met somebody who likes chocolate this way. So I doubt that anybody has bothered to make a recipe for it.

I don't think you can get a more definite answer, as it's generally impossible to prove the nonexistence of something. I can't promise you that nobody has ever made a recipe for it. But the probability is about the same as somebody having created a recipe for an unpeeled banana, and much for the same reasons.

I heard of chocolate bars which are eaten in hotter regions (by soldiers?). The chocolate is brittle, dry and doesn't melt in the mouth. To prevent the eater trying to let the piece of chocolate melt in the mouth the chocolate should be mixed with ingredients that should be chewed anyway - like nuts. I assume that this kind of chocolate is not tempered.

I can't find a link to this claim anymore. The only thing I can remember is that this information was provided by Ritter Sport, a German brand of chocolate.

Close enough? United States military chocolate.

Its ingredients were chocolate, sugar, oat flour, cacao fat, skim milk powder, and artificial flavoring. Chocolate manufacturing equipment was built to move the flowing mixture of liquid chocolate and oat flour into preset molds. However, the temperature-resistant formula of chocolate became a gooey paste that would not flow at any temperature.

Chief chemist Hinkle was forced to develop entirely new production methods to produce the bars. Each four-ounce portion had to be kneaded, weighed, and pressed into a mold by hand. The end result was an extremely hard block of dark brown chocolate that would crumble with some effort and was heat-resistant to 120 °F (49 °C).

I found a use! If you melt out all of the seeds and then don't agitate the chocolate at all while it's cooling, you end up with a ganache-like paste that works pretty well for making vegan, allergen-free truffles and spreads.

Edit: And another! When setting chocolate bark I tried using untempered chocolate that would solidify into streaky, layered flakes. The texture ended up working really well with the nuts and dried fruit.

A characteristic of untempered chocolate is that it melts more easily. That's generally considered bad--if you pick it up in your hand, your hand gets sticky and the surface of the chocolate gets marred. But it also means that you get stronger flavor more quickly when you put it in your mouth. If you are designing a confection where that is the intent, untempered chocolate is preferable.

For example, for chocolate covered strawberries, tempered chocolate results in a crunchy shell, and upon biting in the immediate effect is a crunch and hard unmelted chocolate chunks with a burst of strawberry flavor from the wet strawberry flesh. As the chocolate warms in your mouth, the chocolate melts and that flavor comes in, and the chocolate lingers if you let it melt slowly. On the hand, if you chew quickly and the berry is cold, you can grind up the chocolate without melting it much and even swallow it before it has melted and released its flavor.

With untempered chocolate, there's less of a crunch when you bite in. The chocolate and strawberry tastes hit immediately and pretty much simultaneously. The chocolate melts away more quickly, making the flavor linger a little less. I prefer that effect taste-wise. On the other hand, the strawberries are messier to handle, as the chocolate melts in your hand, and perhaps on your chin. And the snob appeal is less because you don't get to show off your skill and tempering and some people might assume the lack of tempering was due to lack of skill rather than taste.

If you have never tried it, take a chocolate bar you like and break it in half. Melt and re-cool one half. Then try eating a piece from the original tempered half and a piece of the untempered half. You will experience more immediate flavor from the untempered piece, but it will melt and dissipate sooner, so you get to prolong the enjoyment longer with the tempered piece.

A confident confectioner should understand the difference between those experiences and choose appropriately for the experience intended, rather than always using tempered chocolate based on the snobbery factor.

Untempered chocolate is superior IMO in homemade chocolate chip ice cream. Untemper,spread on parcment paper cookie sheet, freeze, break into bits, keep frozen and add to ice cream in the last 3-5 minutes of the churn. The flavor combo when eating is so much better due to the quick melting of the chocolat as you eat the ice cream rather than less flavorful crunchy bits.