I have absolutely no idea how active this group is - I just made a profile here as I love the retro look of this site.
Still, thought it might be fun to list our favorite story or episode by Doctor, as difficult as that is. If you've not seen a Doctor, just skip over them.
First (William Hartnell) - The Keys of Marinus or The Ark
Second (Patrick Troughton) - The Macra Terror or The Mind Robber
Third (Jon Pertwee) - The Time Warrior or The Curse of Peladon
Fourth (Tom Baker) - City of Death, The Brain of Morbius, The Horns of Nimon, or The Sun Makers
Fifth (Peter Davison) - Kinda or Mawdryn Undead
Sixth (Colin Baker) - Trial of a Time Lord (specifically Terror of the Vervoids) or Vengeance on Varos
Seventh (Sylvester McCoy) - The Curse of Fenric or The Happiness Patrol
Eighth (Paul McGann) - "The Night of the Doctor" (which I found better, despite it's short running-time, than the 1996 movie)
Ninth (Christopher Eccleston) - "The Empty Child"/"The Doctor Dances" or "Dalek"
Tenth (David Tennant) - "Human Nature"/"The Family of Blood" or "The Girl in the Fireplace"
Eleventh (Matt Smith) - "A Christmas Carol" or "Amy's Choice"
Twelfth (Peter Capaldi) - "World Enough and Time"/"The Doctor Falls" or "Mummy on the Orient Express"
Thirteenth (Jodie Whittaker) - "The Power of the Doctor"