Jim'll Fix It was a long-running BBC television series hosted by Jimmy Savile. Children from the public would write in to the host requesting that "fix" something for them, to make some wish come true.
The Doctor Who mini-episode "A Fix with Sontarans" featured Colin Baker as the Sixth Doctor came about as a request by a fan named Gareth Jenkins who had his own child-sized version of the Doctor's costume. (A different Gareth Jenkins works with Big Finish Production.)
Other relevant segments
- The Fourth Doctor (Tom Baker) appeared, in-character, during the 1970s, where he handed out torn-off scraps of scarves to some girl visitors to the studio.
- Veteran movie actor Peter Cushing (who played Dr. Who in Doctor Who and the Daleks and Daleks' Invasion Earth 2150 A.D., wrote to the show in 1986 to ask if a variety of rose could be named after his late wife.
- Numerous viewers have reportedly written in wishing for a TARDIS of their own or to ride in the Doctor's.