I couldn't pick among the various dogs we've had, each one someone else's reject, but the one with whom I will always have the closest bond is our little basenji, Rascal, who is now 15 years old. He was about 6-7 months old when we got him, and he had been mistreated by someone. He was terrified of everything, and it took months of coaxing and coddling to bring him around. He became my "baby" and the most endearing little dog you could ever want to know. At 15, he's lost his hearing almost completely, and doesn't see too well, but he still has a lot of zest for living and loves his walks. Because of him we adopted a couple of other second-hand basenjis, who lived to 14 and 17.
I would not recommend basenjis to anyone who doesn't have a lot of time and patience. They're not high strung, but they're very active, and they need to find good ways to spend that energy or they will find bad ways. They are pack animals in a way that no other dogs I have known have been. One alone seems very unhappy and will tend to be troublesome. They evolved in Africa among people who have always kept them outdoors where they could hang together, and they are bred to flush small, quick game. Many of them will never lose the desire to catch anything that will run, and they are extremely speedy. They have terrorized neighbors's cats who came into our yard (exactly once each!). Our three once caught a squirrel. Of course that happened when the DH was out of town and I was left to deal with body parts. They've been harmless to our own cats, but the cats tend to avoid them on principle.
Here's a recent picture of Rascal. He usually has his tail in a tight curl, but sometimes it unfurls when he's very relaxed or asleep. Rascal's current dog pal is Gracie, our sweet little cock-a-poo, and she deserves a lot of mention, too, but I'll save that for another time.