As I'm sure my fellow Oz fans know, L. Frank Baum's tenth Oz book, Rinkitink in Oz, wasn't originally written as an Oz book at all. It was an unrelated fantasy called King Rinkitink, which was never published for some reason, so Baum decided to rework it into an Oz story. Unfortunately, this meant changing the original ending to one where Dorothy and the Wizard of Oz force the Nome King into giving up his prisoners. I can't help but wonder if the earlier ending was more satisfying. Regardless, since it wasn't originally an Oz book, it focuses primarily on new characters and settings. Our three main protagonists are:
Prince Inga - The hero of the book is the young Prince of Pingaree, son of King Kitticut and Queen Garee. He is a serious, studious individual, who luckily decides to read in a tree during a raid by pirates from Regos and Coregos, hence avoiding being kidnapped with his parents and countrymen. He then sets out with his two companions to save his people, with some magical assistance in the form of three pearls that had been given to his ancestor by the Queen of the Mermaids. The blue pearl provides its bearer with great strength, the pink one protects someone from all harm, and the white one gives advice. Seems like they'd make things too easy, doesn't it? Well, sort of, but Inga still faces some challenges on Regos and Coregos, as well as in the Nome Kingdom. Sherwood Smith uses him again in Trouble Under Oz, which takes place some decades later, yet Inga doesn't appear to be any older. I guess we really don't know how quickly Pingarians age, but they're presumably not immortal, or Inga wouldn't have to devote so much time to studying how to be a ruler.
King Rinkitink - The foolish but affable monarch of the Kingdom of Rinkitink, the main trading partner of Pingaree, is largely a comic relief character. He much prefers making jokes and composing songs to ruling, so he sneaks out of his own kingdom to visit Pingaree, and only avoids being captured by the pirates himself because he falls down a well. He serves as a sometimes frustrating but always loyal companion to Inga. While Rinkitink itself is his only major role, he's mentioned in Pirates, and makes a visit to the Emerald City in Lucky Bucky. Moving into apocryphal territory, Red Reera the Yookoohoo and the Enchanted Easter Eggs of Oz has the king marrying his childhood sweetheart Trinket, apparently after a rather long delay.
Bilbil - King Rinkitink's constant companion is a grumpy old goat who, unlike most of the animals of Rinkitink and Pingaree, is capable of talking. He constantly argues with the king, and criticizes his foolishness. Bilbil turns out to have a secret past, actually being Prince Bobo of Boboland under an enchantment by an evil sorcerer. Glinda and the Wizard of Oz restore his true form, and he decides to remain at Rinkitink's court, although there's still the possibility of a return to his old homeland at some point. I tried to cover this in my own not-very-good story "Return to Boboland," and I think David Hulan's The Magic Carpet of Oz covers much the same territory, although I haven't yet had the chance to read that one. Considering that he's one of the few available princes in the area, I like the idea that he might have married Princess Fluff of Noland, who is said to have gotten together with a prince at the end of Queen Zixi of Ix. Then again, I guess it could be Inga who marries Fluff, although I personally like the idea of his eventually marrying Zella, the Regosian peasant girl he befriends during Rinkitink.
I believe Rinkitink was the sixth Oz book I read, and I found it a bit disappointing that it really had little to do with Oz. On rereading it while knowing that it was primarily a non-Oz fantasy, however, I found that I liked it considerably better.