Introduction
This article is, of course, greatly influenced by my preferences, and subjective to some extent. However, all of these breeds are functional and well-loved, so I can't be far wrong.
3.
Araucanas are a south american layer breed with blue eggs and neck ruffs. They also commonly have no rump. Besides these quirky features, they are good foragers and are very hardy once they're adults, but the hatch rate is low and the chick mortality rate is high, so it is hard to propagate them. If you are a beginner in this whole chicken thing, and you want to raise birds from chick-hood, this breed is probably a good choice.
2.
Bantam Silkies are cute little fluff-balls that lay lots of eggs, go broody often, have many babies, forage well, and also happen to be chickens. They can apparently be delicate, especially early on, but I haven't had any problems with that. They're an ancient, functional, breed, and are beautiful and entertaining as well.
1.
Freedom Rangers are probably the best all-around chickens I've found. They lay well, get meaty, and occasionally go broody. Better yet, they are meant to free range, and will eat almost literally everything in sight, so one needs to worry about feeding them too much, rather than starving them. These birds are definitely free rangers, as their name suggests. If you keep them locked up, they get bad feet and bloated, and aren't much use to anyone. But if they're allowed to run free, they make nests, roam over about 3 acres, and leave eggs everywhere. (This is not to say that they'll do badly with any form of confinement. A chicken tractor would be just fine.) On top of it all, they're beautiful and have comically distinctive personalities.
End Notes
In general, as you can see, I prefer resilient, functional chicken breeds that are good free rangers and go broody. I would love to hear what other people say in the comments.