Captain Jack Harkness
What character could be important enough in only a handful of episodes to necessitate a full spin-off? Captain Jack Harkness, of course! Jack Harkness was first introduced in series one as a futuristic conman. However, he quickly befriended the Doctor and Rose and was given immortality by the Bad Wolf after being killed by the Daleks in the finale.
From here, most of Jack's story is told in the spin-off series Torchwood, leading up to his return in Series 3 to help stop the Master. He also returned in Series 4 when the Doctor's many companions gathered to stop the Daleks. After that (and a brief appearance in "The End of Time: Part 2"), he returned in Series 12 to interact with the Thirteenth Doctor and her companions.
Altogether, Captain Jack Harkness appeared in 13 episodes of Doctor Who, during which fans could expect a lot of flirting, some rivalry with the Doctor, and steadfast loyalty to the Time Lord. He is usually an upbeat, enjoyable character in Doctor Who, although his darker side was evident in Torchwood, where he appeared in all 41 episodes. Ultimately, Jack shows another side of immortality, displaying a moral flexibility that suggests why the Doctor might see him as a failure.