Any self-respecting sci-fi nerd has built up an intrinsic need to see ALL of whatever nerd-thing currently has their attention. It's understandable, then, that a lot of us had difficulty wrapping our heads around a show with this much back-story. How could we possibly start following a show fifty years in?
Of course, now that it's gained a foothold in America, we are finally catching on to the fact that the whole point of the show is that it has too much back-story. Even if you've watched every single episode ever made (which is impossible, apparently; a bunch of them are lost), you're still not getting all of the doctor's history. The show basks in it's inscrutableness. Newcomers and longtime fans alike can swim in the exact same water, and we all have a grand time together.
As a relative newcomer, I find myself wishing I had been following the show back when it was a bit more obscure... but I'm having a lot of fun being one of the thousands of Americans jumping onto this particular bandwagon. There aren't a ton of classic Who episodes available to us just yet, but there are a few, and I've been making it my business to watch as much of it as I can in an effort to catch up with the old hands.
While I'd LOVE to try to make some observations about which doctors are cooler than others (10 > 4 > 2 > 5 > 9 > 3 > 7 > 11 > 1 > 6 > 8), there isn't much I can say that hasn't already been said. Instead, to celebrate my finally having watched all the Doctor Who available to me without buying sketchy VHS bootlegs from England, I thought I'd make a list of my favorite companions. It might seem kind of pointless, but one thing I've noticed in these marathon sessions, is that the companions make or break the show. It's basically through them that we come to know the Doctor. Sometimes, the companions are great because of the way their relationship with their particular Doctor makes him a better character, other times, they're awesome all by themselves. Trying to come up with some system for rating these companions against each other has proved more or less fruitless because their Doctors were so different, so instead, I'll just put them in alphabetically.
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Ace
Ace is very specifically the seventh doctor's companion, which puts her squarely in the late eighties. He was a small guy wearing a knitted vest and carrying an umbrella, and she was a feisty teenaged girl with a shady background that carried a backpack of explosives that she fired with a slingshot and beat the paint off a Dalek with a bat. She let the show exist in a post Arnold Schwarzenegger world while allowing the Doctor to stay decidedly intellectual. It was a fantastic pairing, and while Ace is clearly a product of her time, she was so fantastically spunky you can't help but want to see her come back.
Adric
Adric was a product of the Fourth Doctor's time in the TARDIS, which brought us a leotard-wearing wild women, a Time-Lady, and a robot dog. By that standard, a teenaged mathematical genius refugee from an alternate dimension with advanced healing doesn't seem that farfetched. Once the Doctor regenerated into his fifth appearance, Adric's seeming arrogance somehow morphed when paired with the Doctor's new, earnestly sweet personality. It became a desire for approval. You had this sense of the two of them developing a relationship you hadn't seen the Doctor develop before; he was like a big brother. Adric is most famous for being the only companion to have died, and his death profoundly affected the Doctor, so much so that his name was the last word the Doctor said before he regenerated.
Amy & Rory Pond
I may be cheating a little bit by including them together, but I don't think anyone can really fault me for this. As the companions of the 11th Doctor, these have been THE companions for most American newcomers to the show, but they stand out among all the companions because of their unique relationship to the Doctor and to each other. Amy's bond with the Doctor stretches out over her entire life, and because Rory adores her so much, that means that the Doctor is bonded to him as well. There's something really special about that. Plus... you know.... there's his in-laws, so there's that.
The Brigadier
Brigadier Alistair Gordon Lethbridge-Stewart was the long-time leader of the United Nations Intelligence Taskforce, and met the 2nd Doctor in that capacity before the 3rd Doctor served his exile on Earth as the science advisor to UNIT, ostensibly under the Brigadier's authority. For years, the Brigadier suffered constantly being one-upped by the Doctor, always out of his element, and never once has he ever lost his cool. He approached every situation with a uncomplicated military focus, providing the Doctor with constant and unerring support. The Brigadier has appeared again and again across the series. The character was eventually knighted, and was clear to introduce himself as Brigadier Sir Alistair Gordon Lethbridge-Stewart. Even long after his retirement, he continued to help the Doctor and his companions. 
Clara Oswald
I totally admit I may be jumping the gun on this one, but honestly, in all my Doctor Who marathoning, I haven't seen a single companion leap off the screen like Clara. She obviously has the benefit of being not only the Doctor's companion but also the big mystery for the season, so she gets a lot of really awesome moments, but even without that, she absolutely crackles. She gets to do so much and BE so much. She's whip-smart and feisty, compassionate and cheeky, playful and beautiful... and maybe the most important; she gives the Doctor more of a run-around than any other companion. Ever. Out of all of them. That is absolutely fantastic.
Jack Harkness
I honestly can't imagine why I could possibly need to explain this. While he brings both a healthy dose of swashbuckly awesomeness to the TARDIS (to say nothing of the notable uptick in hanky-panky), There's something unmistakeably special about Captain Jack. He's a uniquely fantastic character, and it's to the Doctor's credit that he can count heroic, immortal ex-con adventurers from the future among his companions.
Jamie McCrimmon
Here's a fun fact about Jaime; He was never written to be a companion. He was a highland piper in Scotland from the 18th century that met the 2nd Doctor in a series called "The Highlander". The writers liked him so much, he joined the doctor at the end of the episode. He went on to be the longest running companion of all time. Jamie and the Doctor were a fantastic comedy team, they would riff back and forth through adventure after adventure. He was a brave and loyal friend; always completely out of his element, but sharp enough to, if not work OUT the high technology, then at least work around it. The character is referenced constantly, throughout the entire series. He might be one of the few times when the companion was as much the star of the show as the Doctor himself.
K-9
I honestly challenge a person to not find K-9 absolutely charming. Not only does he have a cool laser, and a chipper little voice that is incredibly helpful with all sorts of useful facts... not only does he have all the inherent coolness of being a robot sidekick, which is pretty cool all by itself, but he's also the Doctor's dog! What relationship could possibly be simpler and more likeable than a dog and it's master? Of all the companions to part ways with the Doctor, he might have been the only one that you kind of worried about, because the idea of a dog being separated from their master is never a wholly pleasant one under any circumstances, so seeing him happily living with Sarah Jane just filled me with joy.
Leela
Leela was part of the Sevateem tribe, descendants of a survey team that crashed on a distant planet in the far future. She barged into the TARDIS and demanded that he take her with him. She is clearly designed to be a sort of Eliza Doolittle character, a wild girl for him to slowly 'tame' over time, but of course, it didn't exactly work out that way. Leela was a little too badass, and her trademark leather leotards proved to be a little too popular with the dads watching the show. You can't possibly not include Leela; there have been other warrior characters in the TARDIS, but Leela stands out as one of the great female television heroes like Wonder Woman or Charlie's Angels. They all had an element of cheesecake to them, but they used it to really kick butt and enhance the possibilities for women as action characters. Doctor Who got to be a part of that because of Leela.
Peri Brown
The 5th Doctor had a very crowded TARDIS. He never had less than two companions at any time, and often had more. Peri was one of his last companions, and was the only one to remain after the regeneration of the 6th Doctor... who immediately tried to kill her. The 5th Doctor had been such a gentle, generous soul, and he and Peri had a very dear friendship, that this sudden change was almost as jarring to us as it surely was for her. It might even be safe to say that it put a lot of viewers off the 6th Doctor right out of the gate. But the fact was, the 6th Doctor was a very complicated incarnation, who had been traumatized in his regeneration, and had dense layers of anger and frustration built up around him. He was a difficult Doctor to sympathize with, and it took a very special companion to help us do it. Peri saw that her friend was troubled and stood by him. She chose to see the good person under the ridiculous patchwork jacket, and it went a long way toward helping us see him too. Peri gets written off a lot because they tended to dress her to show off her body, but the fact is, she's probably the main reason that the show continued during this period.
Rose Tyler
I think it's safe to say that modern Dr. Who fans owe quite a bit to Rose. When the new series started, there was a clear new focus on the relationship between The Doctor and his companion. It was pretty important that they find just the right person to share the TARDIS. Rose was more independent, more capable, more dynamic than so many of the companions before her had been. There have been other companions that seemed to have feelings for the Doctor, but the relationship that grew between these two was certainly unique. You really did watch them grow to love each other, and to have the Doctor continue on without her (even after we discovered that she wasn't going to have to carry on without him) was perhaps the most painful reminder of his endless loneliness.
Sarah Jane Smith
Sarah Jane started as the 3nd Doctor's companion, but went on to travel with the 4th for many years. Jaime has been in more actual episodes, The Brigadier has been a recurring character for longer, and Tegan (the 5th Doctor's displaced air stewardess who didn't want to be there) was there for more seasons, but when you add in Sarah's appearance in modern Doctor Who, her own spin-off series the Sarah Jane Chronicles, and the pilot for K-9 & Company she's actually put in more hours than any other companion. The reason is pretty simple: she is widely regarded as the most beloved companion ever. She was funny, she was brave, she was feisty, and strong, and determined... almost the equal to the Doctor himself. The extraordinary thing about that is that she managed to do it in a time when it just wasn't done. Women just didn't get to be heroic, they were victims or they were decorations. She brought a can-do spirit, a brightness, and an overall aura of being a real, wonderful person that we all wanted to call our friend. Almost every companion that followed owes a least part of who they are to Sarah Jane.
Susan Foreman
The very first companion of the 1st Doctor, Susan has one incredibly unique quality that makes her unique among all the companions that have come since: she was the Doctor's grandaughter. This meant that she herself was a Time-Lord, and implied quite a bit about the Doctor's life before he stole the TARDIS: He had been a father, and a grandfather. Susan was adorable and endearing, and yet had an other-worldliness about her, something that made her feel apart from humanity. There has been another Time-Lord companion in Romana, but Susan's relationship with the Doctor is obviously completely unprecedented. The existence of Susan does more to effect the story of the Doctor than any other companion there has ever been.
Vislor Turlough
The 5th Doctor had quite a few companions, but Vislor didn't only stand out among them, bit among all the companions there have ever been. He was an assassin, and he had been tasked with killing the Doctor. The idea of a character like that existing in the Doctor Who universe is fascinating all by itself, but to have him falling in with the crew of the TARDIS made him absolutely fascinating. Then, of course, there was his incredibly striking look. He was tall, slender, and elegant, somehow coming across as incredibly dangerous. I'm not sure if I'm the only person that felt like there was a vague notion that he might have been gay, but whether he was or not, he was nonetheless an absolutely striking character. It was such a unique energy to have on the TARDIS, and one that was really special.
Zoe Heriot
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While Sarah Jane was really the first truly modern female companion for whom the role of "confident, smart, capable woman" didn't seem more or less painted on, Zoe might have been the first companion to present the idea at all. Zoe was a catsuited, kung-fu fighting astrophysicist with a photographic memory. She was from the distant future, and was so incredibly bright she often outsmarted the 2nd Doctor. There have been spunky, sexy companions, tough, resourceful companions, and brilliant companions; but was the very first that was all of those at once.
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There have been SO many companions, and while these are my favorites, there have been so many stand out moments of so many characters. Even after all this marathon viewing, I still have quite a bit to go. Classic Doctor Who is still a tricky thing to watch in it's entirety, and I'm sure I have quite a few moments of falling in love with other companions in front of me. Even as Doctor Who moves on into a new era, going back to watch these classics is going to be a unique joy.
