Rose Tyler and the Ninth Doctor
The development of their relationship

Some fans will tell you there is nothing romantic between them. I say those fans aren't paying attention. Or rather, they're only paying attention to the plot, the dialogue, and the technobabble, and not to the characters' faces, gestures, and tone of voice. I've even had one fan tell me that's just acting; but really, what else are the characters except what the actors and directors make them? Just lines on a page, pixels on a screen.

Christopher Eccleston, Billie Piper, and several people of the writer, director, or producer persuasion (I'll have to watch the special features again to get names) say that the Doctor and Rose are in love.

Now, the webmaster of the Whoniverse did make a valid point to me: the intentions of the creators are no longer relevant once their creation is done. At that point, we are free to interpret it as we see fit.

So here is my interpretation. You can watch them fall in love over the course of the series, and here are the details.

Rose
Rose: What about you? Are you alien?
Doctor: Yes. Is that all right?
Rose: Yeah.
Notice the way she sighs and smiles. She was playful with him before (hitting him with the plastic arm, almost nudging him while they were walking), but now, she's rather attracted. Unfortunately, she then remembers Mickey.
(deciding whether to travel with him) She's seriously considering it; you can see the excitement on her face. But between his admission that it's dangerous, and Mickey grabbing her, she says no. And then she sees that he's sad: human emotion, though he's been acting so alien (ex. forgetting Mickey twice). She almost reconsiders.
Even when the Tardis is gone, part of her is sorry she said no. There's a look of loss on her face even as she gets up and starts walking away with Mickey. She glances back at the empty space. And when she hears it returning, she whirls around immediately. He's offered her a second chance, and she can't refuse.

The End of the World
Rose: The aliens are so alien. You look at them, and they're alien. The unspoken thought following that line: "And you're one of them." Rose now sees the Doctor as something very different from herself, and therefore places him in that mental box women have labeled "Not a romantic prospect."
Rose: I'm fine. Top of the world. Despite having just counted him out of her love life, she is picking up his habit of making jokes.
(the whole conversation about going for chips) This is when they first begin to connect. She even flirts a little ("What sort of date are you?" and bumping into him with her shoulder), though it's probably completely innocent.

The Unquiet Dead
Doctor: You look beautiful! Considering.
Rose: What?
Doctor: That you're human.
Afraid to let on how much he likes Rose, the Doctor closes the lid on the aforementioned box.
Many fans are inclined to take his words at face value, but that is a big mistake. When he said she was beautiful, his eyes were wide with surprise. It was a sincere comment. It's the "considering" that is insincere; it's a cover-up.
Rose: It happens once, just once, and then it's gone. It's finished. Except for you. You can see days that were lost 100,000 sunsets ago. No wonder you never stay still.
Doctor: Not a bad life?
Rose: Better with two.
Now she's REALLY fascinated with him, and with this new life.
The funny thing is, when a woman has decided a man is not a potential date, she stops guarding her feelings towards him. He's a friend, and you can have all the friends you want, so he's not a threat to any other relationships. He's not even a threat to your own freedom or independence or sense of self. And so maybe, just maybe, you start to fall in love with him, and you don't even notice, because you simply aren't thinking about how you feel. He's a friend.
In the commentary on the DVD, someone (I couldn't tell which voice) suggests that she is "platonically in love" with him in this episode. But of course, that's one of those intentions I should ignore, isn't it!
Doctor: I'm so glad I met you!
Rose: Me too!
If he weren't such a gentleman, and she hadn't already discounted him in that light, and Dickens hadn't returned with a solution right at that moment, they just might have kissed.

Aliens of London
Policeman: When you say "companion"... is this a sexual relationship?
Rose and the Doctor: No.
Look at their faces. Listen to their tones of voice. The Doctor is just being innocent and answering the question, but Rose is also saying, "Don't be ridiculous!"
Mickey: Some boyfriend he turned out to be! Although Jabe was the first one to guess at romantic relationship, I think she was just playing it safe, making sure the Doctor was available before she really got to flirting. But Mickey really believes it.

World War Three
Jackie: Just answer me this: Is my daughter safe?
Rose: I'm fine.
Jackie: Is she safe? Will she always be safe? Can you promise me that?
Jackie has placed him in a protective, almost fatherly role. He doesn't refuse it; he's been saying all along, whenever Rose is in danger, that it's his fault. It's also a fairly common way he's related to his companions in the past (or so we all assume).
Doctor: You don't even know what it is, you'd just let me? This is it. This is the moment that he falls completely head-over-heels, no-turning-back, in love with her. Just look at his face!
Doctor: I could save the world but lose you. He's even trying to tell her, but I think maybe she just sees him asking her permission to take the risk.
Jackie: I have no choice, since you're infatuated.
Rose: I'm not infatuated.
Jackie sees it: Rose is in love. But Rose doesn't see it, because she has already assumed it's impossible.

Dalek
Dalek: What use are emotions if you will not save the woman you love? I think this quote speaks for itself.
Doctor: He's a bit pretty.
Rose: I hadn't noticed.
Doctor: On your own head then.
Having exhausted the logical reasons not to invite Adam along, the Doctor lets slip a hint that he is jealous. Rose says it's not a worry, so he agrees to let him join them.

The Long Game
Adam: You stick with the Doctor. You'd rather be with him. It'll take a better man than me to get between you two. Adam sees it.
Doctor: That's her gone. Adam's given up. Looks like it's just you and me.
Rose: Yep.
Doctor: Good.
Rose: Yep.
They're smiling at each other, and just as the door of the lift is closing, they hold hands.

Father's Day
Pete Tyler: Listen, don't worry about him. Couples have rows all the time. Pete sees it.
Rose: We're not a couple. Why does everyone think we're a couple? (sighs) I think he left me. So, that wouldn't be a mixed signal then, would it?
(when the Doctor is running toward the church) See how happy she looks that he's coming back? Less than an hour, and she missed him.
(no specific citation) Things are thrown off again in this episode, because he is protecting her. Note how he puts his hand on her cheek exactly the way her father did. It is very clear how much he cares for her, though perhaps a little fuzzy in what way.

The Empty Child
Jack: You sort-of have a boyfriend named Mickey Smith, but you consider yourself to be footloose and fancy-free. Only the exact word you use is "available." Oh, and another one: "very." Just in case we were uncertain as to whether Rose consciously thought of the Doctor in a romantic light...

The Doctor Dances
Doctor: You just assume I don't... dance. This is when he realizes that she's been assuming they couldn't have a romantic relationship, and this is his way of telling her that a) they can and b) he wants to. Notice how she immediately starts to flirt with him.
Jack: Most people notice when they've been teleported. You two are so sweet. Once they finally stopped worrying about their surroundings and focused on each other, they didn't notice much else!
Rose: I think Jack would like this dance.
Doctor: I'm sure he would, Rose. I'm absolutely certain of it. But who with?
It's not the lines I'm citing so much as the dancing that follows, and the way Jack looks at them. He knows, in that moment, that he hasn't got a chance of getting involved with either of them.
And I think there's some innuendo in the dip.

Boom Town
Mickey: Do you have to go and tell him?
Rose: It's none of his business.
Jack (inside the Tardis): So, what's on?
Doctor: Nothing. Just...
Unspoken line: "Just watching the woman I love walk away with someone else. Not like I'm jealous or anything."
Rose: Midnight, right, we walk under these waves, 100 feet tall and made of ice.
Mickey: I'm going out with Trisha Delaney.
Mickey sees what a romantic planet that must have been, and he tries to make Rose jealous in return.
Mickey: It's always gonna be the Doctor. It's never me! He knows he's lost her, even if she hasn't come to terms with it yet.
Doctor: How's Mickey?
Rose: He's fine. He's gone.
Doctor: Do you want to go and look for him? We'll wait.
What's most relevant here is the look the Doctor exchanges with Jack. They both know he doesn't want her to go look for Mickey, but neither is willing to say it out loud (the Doctor because he's being noble and trying to hold back and let her decide, and Jack because he knows it's not his place to interfere).

Bad Wolf
Doctor: We were together. We were laughing, and then there was this light, this white light coming through the walls, and then I woke up here. And what he sees in that memory is Rose reaching for him as the transmat beam pulls them apart.
(when the Anne Droid shoots her) 1. She was protecting him by getting in its path.
2. The look on his face as he touches the ashes, and the guards move in... That's not just sorrow; it's not even grief. What it is, is emptiness. He doesn't know how to live without her.
Doctor: Rose?
Rose: Yes, Doctor?
Doctor: I'm coming to get you.
It's not just that he's defying the Daleks to save her; he'd defy the Daleks for no reason at all. It's the way they smile at each other through the screen.

The Parting of the Ways
Lynda: I didn't want to leave you. Look how jealous Rose is! And later, too, when the Doctor and Lynda are trying to decide whether to hug and settle on just shaking hands.
Doctor: Have a good life. Do that for me, Rose. Have a fantastic life.
Rose: You can't do this to me! Take me back! Take me back!
He's so sad to send her away, but he insists on protecting her.
She would rather die together than leave him, but he didn't let her make that choice, so she's begging the Tardis to do it.
Rose: I want you safe. My Doctor. Protected from the false god. No matter what she told Jackie and Mickey about taking a stand, she really did it for the man she loves. You don't call someone "My (insert name)" unless you love him.
Rose: I can see everything. All that is. All that was. All that ever could be.
Doctor: But that's what I see. All the time. Doesn't it drive you mad?
She is finally able to truly understand him, and I think that's when he decides it's okay to make a move. Until then, he was waiting for her; he wasn't going to ask her to get involved when he thought she didn't really know him. 
Doctor: Come here. I think you need a doctor. (Never mind that Rose had a line in between.)
This one is all tone of voice. Sure, he's decided to save her life at the cost of his own, but he also just really wants to kiss her.
(the way he touches her after he's taken the vortex out of her) His hand on her side, on her neck, stroking her face. Men don't do that to women they're not attracted to; they certainly don't do it to girls who are like a daughter.
I honestly believe he would have kissed her again when she awoke, and might even have made love to her then and there (if she'd gone along with it, and I think she would have), if only he wasn't dying.