The Doctor in popular culture and mythology

By the 21st century, the Doctor had become a significant figure in Earth's popular culture variably as a real-life celebrity, a character in fiction, a myth or legend or alternatively as a total unknown.

As a real person
Most accounts regarding the Doctor seem to portray them as a real person. By the end of the 2000s, the Doctor had made appearances on television in multiple incarnations with the Second Doctor even able to call himself famous during the first part of his exile on Earth in the 20th century. (COMIC: Action in Exile et al.)

During their second incarnation's exile, the Doctor lived in luxury at the Carlton Grange Hotel and was very prolific in the public eye. The Doctor acted as a lecturer for a time at London University. (COMIC: The Brotherhood) He also appeared as a guest on the television show Explain My Mystery after becoming famous for his incredible exploits to help solve the mysteries of the viewers. (COMIC: The Night Walkers)

Rowan Cartwright was easily able to contact him after reading the headline "TIME AND SPACE TRAVELLER RESIDING IN LONDON" in the Daily Record (COMIC: The Mark of Terror) and the Brotherhood were able to land a helicopter on top of the hotel and kidnap him with relative ease. (COMIC: The Brotherhood) After the regeneration into his next incarnation and involvement with UNIT, the Doctor faded into obscurity. (TV: Spearhead from Space et al.)

After a Slitheen spacecraft crashed into Big Ben in March 2006 and landed in the River Thames, the Ninth Doctor was photographed by press with Rose Tyler when they arrived at 10 Downing Street to aid the situation with other alien experts. (TV: Aliens of London) Later that year during the Sycorax invasion of Earth, the Queen's Christmas speech was cancelled in favour of a message from Prime Minister Harriet Jones in which she pleaded for the Doctor's help in dealing with the crisis. (TV: The Christmas Invasion)

In 2008, the Tenth Doctor was named and pictured along with Martha Jones and Captain Jack Harkness on the news by a newsreader because of the fact, under the guise of Prime Minister Harold Saxon, ordered a nationwide hunt for the trio on the grounds of suspected terrorism. (TV: The Sound of Drums)