Your favorite companion: New: Donna. Classic: Romana/Leela/Ace/Jamie
Again, this show has dozens of contenders, you can't really pick just one.
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New Who
Donna Noble
I actually like all of the New Who companions, but Donna is a character I can relate to a bit more than others. She starts off as aimless and bored with the world, but then a single Christmas-special-long encounter with the Doctor changes her view of everything, much like what happened to me with the Wayfinder Experience (the LARP camp I go to). Plus, she has a whole season between first meeting him and actually becoming a companion to let everything sink in. Unlike Rose, who has the same background but is actually my age and therefore should be easier for me to identify with, Donna still retains a sense of low self-worth despite her undeniable awesomeness. Guess who else has self-esteem issues? Yeah. Also, she's the ONLY female companion of New Who who has a completely platonic relationship with the Doctor (okay, there was that one kiss in The Unicorn and the Wasp, but that was a) justified by the situation and b) never mentioned again): Rose was his canon OTP, Martha was in unrequited love with him, and Flesh and Stone ended with Amy wanting to sex him up. In retrospect, Donna's position just seems a lot healthier and it didn't make things complicated between them.
Long story short, I love Donna because she's one of the only characters in anything that I look at and think "I'd wanna BE her." (Except for her eventual fate, of course...)
Honorary Mention: Amy Pond
I won't lie: I was nervous when I first saw pictures of her and thought "oh god, Moffat's actually going to start off with a fanservice companion." And then the season actually started and she wound up being a pretty solid character. Also, there've been many times when she's said something in an episode that makes me go "YES. I WOULD SAY THAT" (most notably "Okay, got my spaceship, got my boys..." in The Vampires of Venice). I would go so far as to say that Amy and Eleven is my new favorite Doctor/companion relationship in the new series (I admit I shipped them for a bit, but then Rory did an outstanding job of proving himself both worthy and AWESOME).
Classic Who
Romana
A very unique foil for the Doctor. Even from The Ribos Operation, I was fascinated by the dynamic they had of the street-smart renegade and the book-smart top-of-her-class. She and the Doctor really completed each other. I loved watching the classy smugness of Mary Tamm in Key to Time gradually give way to the confident Doctor's wife Doctor's equal of Lalla Ward. I have to admit, I think I like Romana II more than Romana I, and not just because Four/Romana II is one of my biggest OTPs: it's more a question of ability. With Romana I, it was made clear that she was new to the kind of lifestyle led by the Doctor and that her learnings wouldn't save her every time (Four even mentions this when they first meet). For me, she gave off a sense that she understood the situation, but wouldn't always be able to act on it reliably. This developed in due time, and by the end of The Armageddon Factor she was able to give the Doctor a proper telling-off about his joking around with the Key to Time giving him God Mode. When City of Death swung 'round, I had a greater sense that Romana knew what she was doing nearly all the time. I'll admit that this might be a less interesting character trait for some people, but considering the companion is the role we're supposed to identify with, it just feels...I dunno...nice and secure to me (if you want to call it that) to have a companion who can match wits with the Doctor like that.
My only major issue with her is the abruptness of her departure. The only hint we get in Warriors' Gate that she's possibly about to leave the TARDIS is this little exchange:
Adric: "Well I must say I quite looking forward to going [to N-Space] with you and the Doctor."
Romana: "What if the Doctor and I went different ways?"
Adric: "...But you wouldn't, would you?"
And that's it. And then her actual goodbye scene is so short, I've actually been meaning to time it someday to see how many seconds it is (because it CAN'T be longer than a minute). Oh Doctor, you got your tenses wrong when you said "she'll be superb": she is superb.
Leela/Ace/Jamie
I'm lumping these guys into one blurb mainly because I've only seen about 3-4 serials of each of them, and thus don't really have enough material to make constructive analyses of why I like them. The big thing they have in common is that they're action-oriented companions. Instead of running from the monster, they're much more likely stab it instead. Or blow it up. Or hit it with a baseball bat.
Day 01 - Your Favourite Quote: Multiple
Day 02 - Your Favourite Classic Series Episode: The Mind Robber/Battlefield
Day 03 - Your Favourite New Series Episode: Girl in the Fireplace/Midnight/Vincent and the Doctor
Day 04 - Your Favourite Doctor: Ten (New), Three (Classic)
Day 05 - Your Favourite Companion
Day 06 - Whatever Tickles Your Fancy
Day 07 - Your Favourite Piece of Music
Day 08 - A Who-Related Photo That Makes You Happy
Day 09 - A Who-Related Photo That Makes You Angry/Sad
Day 10 - A Who-Related Photo That You Took
Day 11 - Your Favorite Season (Classic or New)
Day 12 - Whatever Tickles Your Fancy
Day 13 - Your Favorite Villain
Day 14 - The Villain Who Scared You the Most
Day 15 - Favorite Who-Related Tumblr
Day 16 - Your Favorite Who-Related FanFic
Day 17 - A Piece of Who-Related FanArt
Day 18 - Whatever Tickles Your Fancy
Day 19 - The Scene That Made Your Cry the Most
Day 20 - The Character Who Is Most Like You
Day 21 - Your Doctor Who OTP
Day 22 - A Who-Related Fan-Site
Day 23 - A Who-Related YouTube Video
Day 24 - Whatever Tickles Your Fancy
Day 25 - Favorite Who Actor
Day 26 - Favorite Who Actress
Day 27 - An Episode You Wish Hadn’t Been Made
Day 28 - An Episode Idea You Created Yourself
Day 29 - Who You Think Should Be the Next Doctor
Day 30 - Whatever Tickles Your Fancy