Fan

Fans (short for fanatic) were people who had a strong interest or devotion in something or somebody. Before realising that he was his future incarnation, the Fifth Doctor initially believed that the Tenth Doctor was a fan, suspecting that he was a member of LINDA. (TV: Time Crash) The Whizz Kid was a fan of both the Psychic Circus and Captain Cook. (TV: ''The Greatest Show in the Galaxy)

The Ninth Doctor identified himself as a fan of Charles Dickens. With Dickens unfamiliar with the term, the Doctor clarified that the term was short for "fanatic". (TV: The Unquiet Dead)

Yani Mamora was a fan of Doctor Who? from Gharusa Prime, even becoming a member of the Doctor Who Appreciation Society. When the Ninth Doctor came to Gharusa Prime, she impressed him with her morals. The Doctor recorded a message for her, encouraging her to stop obsessing over him and saying he was her number one fan. (COMIC: Doctormania)

In a parallel universe where the Doctor's life was a TV show, Ally was a fan of the Doctor's adventures. (COMIC: The Girl Who Loved Doctor Who)

Supergirl was a "huge fan of disguises". (WC: Supergirl Meets E.T.)

Invalid sources
A fan of the Doctor attended the Annual Gallifreyan Doctor Who Con where they asked the first six incarnations of the Doctor and Dr. Who about the most evil foe they had encountered over the years. (NOTVALID: Doctor Who? 95)

The Cyber Fan, a variant of Cyberman, asked the Seventh Doctor for his autograph. (NOTVALID: Doctor Who? 204)