18 Seconds
He sees Rose holding her boyfriend, and the TARDIS fades away, leaving her there. "Well," he says, clapping his hands together. "Now what?"
He starts off by going north. Even though 'north' is a somewhat moot point when travelling in the five dimensions, it's still a place to start.
He goes further and further, pushing the TARDIS as far as it wants to go, laughing as it sizzles and shakes and creaks against the strain of it all.
He ends up on a tiny little planet at the far edge of space, where humanoids drink extra strong extra sweet tea and eat greasy breakfasts.
Everywhere's got a North, y'know. And the breakfast is delicious.
He finds a tattered copy of "Who's Who In Galactic Politics, 52.700.1 to 50.662.5" underneath a table leg in a spare room he forgot the TARDIS even had and decides to go and visit every one. He has tea with the last vicar in all of humanity. He has cake with the 257th direct descendant of an obscure monarch of an even obscurer country on an even more obscure planet. He argues with pundits, trades quips with economists, and even gets into a fistfight with a well-respected interplanetary pacifist.
Despite all that, he gets bored halfway through "G'a to Gm" and sticks it back under the table.
He's walking through the streets of London on Christmas Eve when shrill shouting nearly causes him to trip over several shoppers.
"I told you it had to be completely put together! Look at this! The tyres are off, there's no basket --"
"How're y'gonna fit it in the car, then, huh? Can't stick it in the boot w'tyres!"
"And how am I supposed to give this to her then? Christmas bloody Eve, and you can't even get the tyres on! I trusted you, y'know. You said you'd get her a bicycle -- for cheaper than Argos as well -- and you'd get it all set up so that it'd be right next to the tree on Christmas morning, and look at it! Just look at it!"
"You can't get it in the boot with the tyres, Jackie! How many times do I gotta say this?"
"Then you can tell Rose that her good-for-nothing 'Uncle Mike' can't even put a bike together properly! Christmas morning and it's 'Sorry, Rose, couldn't put it in the boot of my Rover with the tyres on'!"
"Jackie!"
"'Guess you'll just have do it yourself, love.' Like everything else in the bloody house!"
He tries to move on, but stops, shakes his head, and stands up. He walks over to where two people are arguing and stands there, watching them. "Pardon me," he says after a few seconds. "Can I help?"
"Who're you then?" shouts the man.
"What do you want?" shouts the woman.
He opens his mouth, then closes it, smiling tightly. "Never mind." He leans down, pulls out his sonic screwdriver, and with three quick clicks, the bicycle (second-hand, bright red) is finished.
"It should fit in the boot now," he says, just before walking away.
He briefly considers going to a place that would have memories of Gallifrey -- a library, or a museum, perhaps. A place with momentos, with artifacts, even the occasional relic.
He has a faint recollection of very large headdresses and dourness and decides he'd probably be miserable anyway.
The banana groves of Villengaard are fantastic -- there's no other way to describe them. Against the burnt-out shell of the factory (the brickwork crushed into the ground to make perfect drainage), there is row after row of trees, swaying gently in the breeze.
He's strolling along the carefully picked out road (cobbled in broken casings of sonic guns), when there's a shout behind him. He turns, and there's a woman, smiling widely, leading a donkey piled down with bananas.
"Doctor!" she says cheerfully. "How nice to see you again!"
He smiles and walks to her, picking her up in a spinning hug that leaves her laughing and dizzy. She reaches into one of the bags on the donkey's back and pulls out a bunch of bananas -- ripe and golden in the sunlight. "First of the harvest," she says, holding them out to him. "You should have them."
He grins and takes them. Together, they walk along the road, while she tells him what has happened as he nods and eats a banana.
He notices that her dress is slightly too tight around the middle. When he points this out, making a joke about eating too much of her own produce, she blushes, and cradles her stomach protectively. When she laughs so much that she starts to cough, he holds her for a bit, patting her back until it passes. He takes her hand, the stump of her index finger still hard against his palm, as they talk, and they swing their arms back and forth.
Even after 55,000 years, the makers of weapons know one thing about production -- if you want something done right, you still get a young woman to do it. Lose a finger, still have four more to continue polishing casings with. Constantly breathing in fumes and fibres might make you develop a cough, but as long as you don't cough a week's profits down the assembly line, you'll be fine. And if you even think about having children, forget it, because there's no maternity leave, no childcare, and you might as well be dead.
The banana trees rustle in the breeze, and, as he waves goodbye to her, he's still smiling.
He runs into a previous regeneration of himself. He runs into a future regeneration of himself. Both of them make fun of his clothing. Both of them should talk.
He doesn't care about humanity or Earth all that much, but he keeps on coming back. It might have something to do with one of his previous generations. Or maybe it's the food.
Certainly not the weather. But, then again, he did choose an ice age.
Snow builds up against the TARDIS while he sits in a cave, tracing his fingers over still-tacky paintings of bison and bears and men.
He helps the men of the clan kill another bear, and for this honour, they place his hand against the cave wall before covering it in ochre. When he removes his hand, the outline stays on the wall.
The cave wall was damp and cool against his heated skin. The ochre dries, cracking and flaking off. He looks at the handprint on the wall.
He comes back to the cave 17,000 years later. Under floodlights and scientific observation, he places his hand back against the outline.
He sees the Face of Boe shortly before the pregnancy is announced.
Whatever happened was between the two of them.
He wants to hide out in the midst of humanity after that, so he heads for one of the larger, happier colonies, the kind that has smiling children and purring kittens and everything you'd expect from a species that jumped forward into the universe and made it as bright and shining and insane as you would expect from such a creation.
He has a bite to eat from a nearby stall, he buys a drink from a wandering seller, and he watches the masses of humanity go by, smiling and talking and being human.
An old woman walks by, surrounded by what could have only been her great-great-grandchildren and she stops, looking right at him.
He looks back, frowning slightly as she smiles, her eyes gleaming brightly amid the wrinkles.
"Doctor?" she says, a faint note of a London accent to her voice.
His frown deepens a bit more, and then, suddenly, the realisation kicks in. His eyes widen, and a huge smile breaks out across his face. "Rose...?"
Rose just grins.
"Rose!" He stands up, knocking over his drink. "Rose, what're you --" He stops. "Wait a tick. It's..."
"7-nought-3-3-1," she says. "Or...I keep on forgetting the maths. 3324 back on Earth?"
"But you...you should..." He gestures around. "I know there are a number of innovations in the medical field in the first half of the 21st century, but thirteen hundred years is a bit much..."
She laughs. "If I had stayed there, yeah."
He looks at her, and the children, and the smile she has on her face, and it all comes together.
"Oh..."
She smiles even wider.
"Fantastic!"
The TARDIS landed back right where he left it, in that small alley in the centre of London, and he stuck his head out, grinning. "Did I mention it also travels in time?"