The Master
While it doesn't really make sense to call the Master a recurring character most of the time, since they fit in the villain category so well, Missy actually appears several times as an ally for the Doctor. Because of that, and the character's complex relationship with the Doctor, they earn their spot on this list.
The Master is the Doctor's oldest friend and greatest enemy, first appearing on-screen in the Third Doctor's era. In general, the Master's goal is to destroy things and rule over the ruins, although he also frequently causes chaos just to get the Doctor's attention.
In the Classic Era, the Master was portrayed by Roger Delgado, Peter Pratt, Geoffrey Beevers, Anthony Ainley, in a total of 75 episodes. He then appeared in several forms in the 1996 TV movie, before returning in the revival for 25 episodes, played by Derek Jacobi, John Simm, Michelle Gomez, and Sacha Dhawan.
What does it mean when the Master appears in Doctor Who? Usually, there's a high body count with at least one beloved character killed off to show how much of a threat the Master can be. They usually tow the line between scheming and insanity, using their knowledge of the Doctor as a weapon. In the end, though, the Master is always going to be defeated and the Doctor will be sad, both with how far their friend has fallen and what they had to do to stop them.