I haven't updated this in a while, but since the first episode of the Thirteenth Doctor aired this week, why not?
Serials from the original series are identified by their production code and a three digit number of "overall" serial numbers as established by Doctor Who Magazine #407 and used on
Wikipedia. The 2005+ show will only have the three digit numbers.
The Primary List, the actors chosen to be the incumbent Doctor in order they were chosen
Classic Series
1. William Hartnell (1963-1966)
"An Unearthly Child"(A/001)---"The Tenth Planet" (DD/029); reprises the role in "The Three Doctors" (RRR/065, 1973)
Companions:
"Susan Foreman"/Arkytior (Carole Ann Ford) A/001 "An Unearthly Child"---K/010 "The Dalek Invasion of Earth"
Ian Chesterton (William Russell) A/001 "An Unearthly Child"---R/016 "The Chase"
Barbara Wright (Jacqueline Hill)A/001 "An Unearthly Child"---R/016 "The Chase"
Vicki Pallister (Maureen O'brien) L/011--U/020 "The Myth Makers"
Steven Taylor (Peter Purves) R/016 "The Chase"---AA/026 "The Savages"
Katarina (Adrienne Hill) U/020 "The Myth Makers"--V/021 "The Dalek Masterplan"
Sara Kingdom (Jean Marsh) V/021 "The Dalek Masterplan"
Dodo Chaplet (Jackie Lane) W/022 "The Massacre Of St. Bartholomew's Eve"--BB/027 "The War Machines"
Ben Jackson (Micheal Craze) BB/027 "The War Machines"--DD/029 "The Tenth Planet"-->
Polly Wright (Anneke Willis) BB/027 "The War Machines"--DD/029 "The Tenth Planet"-->
2. Patrick Troughdon (1966-1969)
"The Tenth Planet"(DD/029)/"Power of the Daleks"(EE/030)---"The War Games"(ZZ/050). Reprises the role in "The Three Doctors"(RRR/065, 1973), "The Five Doctors"(6K/129, 1983), and "The Two Doctors"(6W/140, 1984)
Companions:
Ben Jackson (Micheal Craze) <--EE/030 "Power of the Daleks"--KK/035 "The Faceless Ones"
Polly Wright (Anneke Willis)<--EE/030 "Power of the Daleks"--KK/035 "The Faceless Ones"
Jamie McCrimmon (Frazier Hines) FF/031 "The Highlanders"--ZZ/050 "The War Games"
Victoria Waterfield (Deborah Watling) LL/036 "Evil of the Daleks"--RR/042 "Fury From The Deep"
Zoe Herriot (Wendy Padbury) SS/043 "The Wheel In Space"--ZZ/050 "The War Games"
Recurring:
Brigadier Alistair Lethbridge-Stewart (Nicholas Courtney) 1:QQ/041 "The Web of Fear"
Sergeant John Benton (John Levene) 1:VV/046 "The Invasion"
3. Jon Pertwee (1970-1974)
"Spearhead from Space"(AAA/051)---"Planet of the Spiders"(ZZZ/074). Reprised the role in "The Five Doctors"(6K/129, 1983)
Companions:
Liz Shaw (Caroline John) AAA/051 "Spearhead From Space"--DDD/054 "Inferno"
Jo Grant (Katy Manning) EEE/055 "Terror Of The Autons"--TTT/069 "The Green Death"
Sarah Jane Smith (Elisabeth Sladen) UUU/070 "The Time Warrior"--ZZZ/074 "Planet of the Spiders"-->
Recurring:
Brigadier Alistair Lethbridge-Stewart (Nicholas Courtney)
Sergeant John Benton (John Levene)
Captain Mike Yates (Richard Franklin) 1: EEE/055 "Terror of the Autons"
4. Tom Baker (1974-1980)
"Planet of the Spiders"(ZZZ/074)/"Robot"(4A/075)---"Logopolis"(5V/115)
Tom is the Doctor most familiar to older American fans--PBS started rerunning his stories in the 1980's. He's the one with the thick curly hair and the super-long scarf. See also the Curator below.
Companions:
Sarah Jane Smith (Elisabeth Sladen) <--4A/075 "Robot"--4N/087 "The Hand of Fear"
Harry Sullivan (Ian Marter) 4A/075 "Robot"--4F/080 "Terror of the Zygons"
Leela (Louise Jameson) 4Q/089 "The Face Of Evil"--4Z/097 "The Invasion of Time"
K9 Mk1 (John Leeson) 4T/093 "The Invisible Enemy"--4Z/097 "The Invasion of Time"
K9 Mk2 (John Leeson/David Brierley) 4Z/097 "The Invasion of Time"--5S/113 "The Warrior's Gate"
Romana 1 (Mary Tamm) 5A/098 "The Ribos Operation"--5F/103 "The Armageddon Factor"
Romana 2 (Lalla Ward)5J/104 "Destiny of the Daleks"--5S/113 "The Warrior's Gate"
Adric (Matthew Waterhouse) 5R/111 "Full Circle--5V/115 "Logopolis"-->
Nyssa (Sarah Sutton) 5T/114 "The Keeper of Traken"--5V/115 "Logopolis"-->
Tegan Jovanka (Janet Fielding) 5V/115 "Logopolis"-->
Recurring:
Brigadier Alistair Lethbridge-Stewart (Nicholas Courtney)
Sergeant John Benton (John Levene)
5. Peter Davison (1980-1984)
"Logopolis"(5V/115)/"Castrovalva"(5Z/116)---"The Caves of Androzani"(6R/135); Reprises the role in "Time Crash"(187.1, 2007)
Companions:
Adric (Matthew Waterhouse) <--5Z/116 "Castrovalva"--6B/121 "Earthshock"
Nyssa (Sarah Sutton) <--5Z/116 "Castrovalva"--6G/126 "Terminus"
Tegan Jovanka (Janet Fielding) <--5Z/116 "Castrovalva"--6P/133 "Resurrection of the Daleks"
Vislor Turlogh (Mark Strickson) 6F/125 "Mawdryn Undead"--6Q/134 "Planet of Fire"
Perpugilliam "Peri" Brown (Nichola Bryant) 6Q/134 "Planet of Fire"--6R/135 "The Caves of Androzani"-->
Recurring:
Brigadier Alistair Lethbridge-Stewart (Nicholas Courtney)
6 Colin Baker (1984-1986)
"The Caves of Androzani"(6R/135)/"The Twin Dilemma"(6S/136)---"Trial Of A Time Lord (The Ultimate Foe)"(7C, 143D)
Companions:
Perpugilliam "Peri" Brown (Nichola Bryant) <--6S/136 "The Twin Dilemma"--7B/143B "Trial Of A Time Lord (Mindwarp)"
Melanie Bush 7C/143C "Trial Of A Time Lord (Terror Of The Vervoids)"--7C/143D "Trial Of A Time :Lord (The Ultimate Foe)"-->
7. Sylvester McCoy (1987-1990, 1996)
"Time and the Rani"(7D/144)--"Survival"(7P/155)
The original show was cancelled in 1990, but McCoy got to reprise the part in the 1996 TV movie (156), and be seen regenerating into his successor.
Companions:
Melanie Bush (Bonnie Langford) <--7D/144 "Time And The Rani"--7G/147 "Dragonfire"
Dorothy "Ace" McClane (Sophie Aldred) 7G/147 "Dragonfire"--7P/155 "Survival"
Recurring:
Brigadier Alistair Lethbridge-Stewart (Nicholas Courtney)
8. Paul McGann (1996)
Only made one onscreen appearance as the incumbent Doctor, the 1996 TV movie (156). Reprises the role in 2013's "Night of the Doctor" (239.1). Has made a large number of audio dramas for Big Finish; because their contract only covered the original show, until 2015 McGann is treated as their incumbent Doctor. He namedrops some of his Big Finish companions in "Night of the Doctor" making at least some elements of the audio dramas provisionally canonical.
Companions:
Dr. Grace Hallowell (Daphne Ashbrook) 156 (TV Movie)
Charlotte "Charlie" Pollard (India Fisher)(Big Finish Audio--mentioned in 239.1 "Night of the Doctor")
C'rizz (Conrad Westermaas)(Big Finish Audio--mentioned in 239.1 "Night of the Doctor")
Lucie Miller (Sheridan Smith)(Big Finish Audio--mentioned in 239.1 "Night of the Doctor")
Tamsin Drew (Niky Wardley)(Big Finish Audio--mentioned in 239.1 "Night of the Doctor")
Molly O'Sullivan (Ruth Bradley)(Big Finish Audio--mentioned in 239.1 "Night of the Doctor")
Revived Series.
(See 8.5 below for John Hurt)
9. Christopher Eccleston (2005)
"Rose"(157)---"The Parting of the Ways"(166)
Per "Night of the Doctor"(239.1), "Day of the Doctor" (240), and "Time of the Doctor" (241), he is actually the Doctor's tenth biological incarnation.
Companions:
Rose Tyler (Billie Piper) 157 "Rose"--166 "The Parting of the Ways"-->
Recurring:
Mickey Smith (Noel Clarke) 1: 157 "Rose"
Adam Mitchel (Bruno Langley) 1: 161 "Dalek"
Jack Harkness (John Barrowman) 1: 164a "The Empty Child"
10. David Tennant (2005-2010)
"The Parting of the Ways"(166)/"The Christmas Invasion"(167)---"The End of Time" (202); Reprises the role in "Day of the Doctor" (240, 2013)
Per "Night of the Doctor"(239.1), "Day of the Doctor" (240), and "Time of the Doctor" (241), he is actually the Doctor's eleventh and twelfth biological incarnations. The creation of the "Meta Crisis Doctor" in "Journey's End"(198) allowed the Tenth Doctor to regenerate without changing his appearance.
Companions:
Rose Tyler (Billie Piper) <--167 "The Christmas Invasion"--177B "Doomsday"
Martha Jones (Freema Agyeman) 179 "Smith and Jones"--187C "The Last of the Time Lords"
Donna Noble (Catharine Tate) 189 "Partners In Crime"--198B "Journey's End"
Recurring:
Mickey Smith (Noel Clarke)
Jack Harkness (John Barrowman)
Melody "River Song" Pond Williams (Alex Kingston) 1: 195a "Silence In the Library"
11. Matt Smith (2010-2013)
"The End of Time"(202)/"Eleventh Hour"(203)---'Time of the Doctor"(241)
Per "Night of the Doctor"(239.1), "Day of the Doctor" (240), and "Time of the Doctor" (241), he is actually the Doctor's thirteenth biological incarnation, the last of his original life cycle. "Time of the Doctor" explains how he gets around that.
Companions:
Amelia "Amy" Pond (Karen Gilen) 203 "Eleventh Hour"--230 "The Angels Take Manhattan"
Rory Williams (Arthur Darvill) 207 "The Vampires of Venice"--230 "The Angels Take Manhattan"
Clara Oswald (Jenna Coleman) 232 "The Bells of St. John"--241 "Time of the Doctor"-->
Recurring:
Melody "River Song" Pond Williams (Alex Kingston)
Madame Vastra (Neve McIntosh) 1: 218 "A Good Man Goes To War"
Jenny Flint (Catrin Stewart) 1: 218 "A Good Man Goes To War"
Strax (Dan Starkey) 1: 218 "A Good Man Goes To War"
12. Peter Capaldi (2013-2017)
"Time of the Doctor"(241)/"Deep Breath"(242)-"Twice Upon A Time" (276)
Per "Night of the Doctor"(239.1), "Day of the Doctor" (240), and "Time of the Doctor" (241), he is the Doctor's fourteenth biological incarnation, and the first regeneration of a new life cycle.
A glimpse of him is seen in "Day of the Doctor"(240), but he officially debuts in "Time of the Doctor"(241)
Companions:
Clara Oswald (Jenna Coleman) <--242 "Deep Breath"--262 "Hell Bent"
Nardole (Matt Lucas) 263 "The Husbands of River Song"--275b "The Doctor Falls"
Pearl Mackie (Bill Potts) 265 "The Pilot"--276 "Twice Upon A Time"
Recurring:
Melody "River Song" Pond Williams (Alex Kingston)
Madame Vastra (Neve McIntosh)
Jenny Flint (Catrin Stewart)
Strax (Dan Starkey)
13. Jodi Whittaker (2017-?)
"Twice Upon A Time" (276)/"The Woman Who Fell To Earth"(277) -->
Per "Night of the Doctor"(239.1), "Day of the Doctor" (240), and "Time of the Doctor" (241), she is the Doctor's fifteenth biological incarnation, and the second regeneration of a new life cycle. She is the first incarnation of the Doctor to be a woman; while regenerative gender-swapping is rare, it is not unknown, and it has been suggested (in fan circles anyway) that the "unnatural" regenerations gained in "Time of the Doctor" may play a role in the Doctor's case.
Companions:
Yasmin Khan (Mandip Gill) "The Woman Who Fell To Earth"(277) -->
Graham O'Brien (Bradley Walsh) "The Woman Who Fell To Earth"(277) -->
Ryan Sinclair (Tosin Cole) "The Woman Who Fell To Earth"(277) -->