You know what, I am decently drunk and this seems like a fun hour to spend, so I will tally up the numbers for our three main contenders, Jamie, The Brigadier and Sarah Jane. I don't think that anybody else come close, so we will work with these three. We will weight Sarah Jane's modern episode as two in order to give fair value to full time. I will go by the number of episodes in the serial (since it is almost all 20th Century Who) and count those episodes as for that character, except where I distinctly remember the character being there for less. There may be an episode or two wherein a companion does not appear due to the script, illness or holiday, but, nevertheless, this should serve as a somewhat rough, but fairly accurate episode count for each companion. (In other words, I am not going to re-watch every single episode to make sure that each companion appeared in them for the absolute accuracy of this exercise.) Also, I will be ignoring "The Brigadier's Corpse's return in Moffat's abomination Death In Heaven since it was not The Brigadier or even his corpse, but an animated Cyberman who may or may not have had The Brigadier's memory.
Jamie McCrimmon:
Second Doctor Period:
The Highlanders = 4 Episodes
The Underwater Menace = 4 Episodes
The Moonbase = 4 Episodes
The Macra Terror = 4 Episodes
The Faceless Ones = 6 Episodes
The Evil of The Daleks = 7 Episodes
The Tomb of The Cybermen: 4 Episodes
The Abominable Snowmen: 6 Episodes
The Ice Warriors: 6 Episodes
The Enemy of The World: 6 Episodes
The Web of Fear: 6 Episodes
Fury from the Deep: 6 Episodes
The Wheel in Space: 6 Episodes
The Dominators: 5 Episodes
The Mind Robber: 5 Episodes
The Invasion: 8 Episodes
The Krotons: 4 Episodes
The Seeds of Death: 6 Episodes
The Space Pirates: 6 Episodes
The War Games: 10 Episodes
Fifth Doctor:
The Cave of Androzoni: (I believe just) 1 Episode
Sixth Doctor:
The Two Doctors: 3 Episodes, but double length, so: 6 Episodes
Total: 120 Episodes.
The Brigadier:
Second Doctor:
The Web of Fear: 6 Episodes (but he came in famously in the still missing Episode 3), so: 4 Episodes
The Invasion: 8 Episodes
Third Doctor:
Spearhead From Space: 4 Episodes
Doctor Who and the Silurians: 7 Episodes
The Ambassadors of Death: 7 Episodes
Inferno: 7 Episodes
Terror of the Autons: 4 Episodes
The Mind of Evil: 6 Episodes
The Claws of Axos: 4 Episodes
The Daemons: 5 Episodes
The Time Monster: 6 Episodes
The Three Doctors: 4 Episodes
The Green Death: 6 Episodes
Invasion of the Dinosaurs: 6 Episodes
Planet of the Spiders: 6 Episodes
Fourth Doctor:
Robot: 4 Episodes
Terror of the Zygons: 4 Episodes
The Masque of Mandragora: 4 Episodes
Fifth Doctor:
Mawdryn Undead: 4 Episodes
The Five Doctors: 1 Episode (triple length), so: 3 Episodes
Seventh Doctor:
Battlefield: 4 Episodes
Total: 111 Episodes
Sarah Jane Smith:
Third Doctor:
The Time Warrior: 4 Episodes
Invasion of the Dinosaurs: 6 Episodes
Death to the Daleks: 4 Episodes
The Monster of Peladon: 6 Episodes
Planet of the Spiders: 6 Episodes
Fourth Doctor:
Robot: 4 Episodes
The Ark in Space: 4 Episodes
The Sontaran Experiment: 2 Episodes
Genesis of the Daleks: 6 Episodes
Revenge of the Cybermen: 4 Episodes
Terror of the Zygons: 4 Episodes
Planet of Evil: 4 Episodes
Pyramids of Mars: 4 Episodes
The Android Invasion: 4 Episodes
The Brain of Morbius: 4 Episodes
The Seeds of Doom: 6 Episodes
The Masque of Mandragora: 4 Episodes
The Hand of Fear: 4 Episodes
Fifth Doctor:
The Five Doctors: 1 Episode (Triple length), so: 3 Episodes
Tenth Doctor:
School Reunion: 1 Episode (Double length), so: 2 Episodes
The Stolen Earth/Journey's End: 2 Episodes (Double length), so: 4 Episodes
Total 89 Episodes
I think that this is all correct (but please feel free to check my math since I have been drinking wine). If it is all reasonably accurate, then my instincts were correct and it is Jamie followed by The Brigadier followed by Sarah Jane.