The Daleks themselves said in "Asylum of the Daleks" that it was known throughout the universe that The Doctor requires companions in his travels. They didn't know why, but they acknowledged that it was so. He's had dozens, maybe hundreds of them in his running through all of time and space. Some last only a brief while, while others stay with him for what seems like ages. Who is the longest serving companion? Well, that depends on how you measure it.
Jamie McCrimmon accompanied the Second Doctor in 113 episodes, far more than any other companion. Frazer Hines happened to be working on the show during one of its busiest times ever and stayed with the program at the request of Patrick Troughton who wished to have Jamie and The Doctor together until the end.
In terms of who has been in the most stories the clear winner is Amy Pond. She was with the Eleventh Doctor in 25 separate television adventures. Rose Tyler is the only companion to travel with a Doctor on every single one of his televised (and in Nine's case prose as well) outings. Jamie comes in second, having missed only "The Power of the Daleks" in Troughton's era.
If you consider how long they were on the show then Tegan Jovankan has the longest tenure, just narrowly beating out Sarah Jane Smith. Janet Fielding held the role for just under three years. Her and Elizabeth Sladen are the only two companion actresses to have featured in four consecutive seasons.
This does of course discount the Brigadier, who remains a rather unique associate of The Doctor that may or may not qualify as a companion in the stricter definition of the term. It is worth noting that Nicholas Courtney has appeared in more episodes than Peter Davison, Colin Baker, Sylvester McCoy, Paul McGann, and Christopher Eccleston, lending some credit to the notion that The Doctor is actually a companion of the Brigadier.
The companion to have spent the longest in terms of years in the narrative with The Doctor is Handles, the Cyberman head that stayed with the Eleventh Doctor on Trenzalore. He stayed with The Doctor for at least 300 straight years. It's possible to say that Rory Williams tops Handles if you consider that from The Doctor's point of view Rory left him only momentarily to guard the Pandorica containing Amy Pond, but from Rory's perspective it was 2,000 years. It all depends on whether the linear duration of the adventure or the duration of The Doctor's attention is the defining factor.
Charley Pollard is the most prolific companion in the Big Finish Audio range, starring in 34 radio plays with the Sixth and Eighth Doctor and beating out Ace with 30... a lead that will likely disappear as Ace continues to make appearances and Charley has not appeared since 2009. If the audio stories are broken down into their four separate episodes that actually puts Charley ahead of both Jamie in terms of number of episodes spent traveling with The Doctor, and well ahead of Amy Pond for adventure count.
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How the Big Finish plays should "count", not to mention novels, short stories, and comics, is extremely dicey, though, and should in general only apply to the original Eighth Doctor companions because he has no regular companions otherwise. If you add in the prose and audio excursions regarding the other television companions then it's likely that Ace, with her long run in the New Adventure novels, prolific Big Finish career, and appearances in the Doctor Who Magazine comics between the show's cancellation in 1989 and the birth of the Eighth Doctor in 1996, is the top contender for the most appearances of a companion across all media.
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Another way to look at the question is to ask how long The Doctor has known someone total. His granddaughter Susan Foreman was obviously his first true companion as we understand the term, yet she was also the first to leave the show. That would by no means be Susan's last adventure, though. She would work with the Eighth Doctor to protect earth from the Daleks near the end of his time with companion Lucie Miller. In that time he met his great-grandson Alex.
Susan's teachers Ian Chesterton and Barbara Wright might have Susan beat in that regard, though. They would be drawn into the world of The Doctor when aliens calling themselves The Hunters of Burning Stone attacked Earth and the Eleventh Doctor came to their aid. After saving the day with their help he attended their wedding, having by then known the couple for over a thousand years from his perspective.
Then again, if The Doctor has any true companion it is the Tardis, who will eventually be with him even after he dies if "The Name of the Doctor" is taken to be inevitable. Like the Brigadier, whose image the broken Tardis once assumed to help the Eighth Doctor fight the evil anti-time gestalt Zagreus, the Tardis belongs in a category all her own.