River Song
River Song is the most prominent recurring character introduced by the modern version of Doctor Who, appearing in 15 episodes so far, as well as multiple audio dramas and novels. She first appeared in the Series 4 two-parter "Silence in the Library" and "The Forest of the Dead." At that point, she was a complete mystery to the Doctor, a figure from his future who knew him better than anyone.
Her secrets were revealed over the course of Series 5 through 7, although there is always the potential for more appearances in the future. The audio dramas have found excuses for her to bump into just about every previous version of the Doctor, so there's no reason she couldn't show up in the future as well, as she did in the 2015 Christmas special. She is the daughter of companions Rory Williams and Amy Pond, the Doctor's wife, and a renowned criminal-slash-archaeologist throughout the universe.
When River appears in an episode, it usually goes one of two ways. Either she knows more than the Doctor, in which case she helps him, teases him, and generally makes him desperate for more information, or he knows more than she does, in which case she's usually trying to kill him. Either way, they have excellent chemistry, and audiences get to see the Doctor out of his element, something very few other characters can manage.