And here's some back story to get us up to the present. The story proper, chronicling the "in-betweens" of the Doctor and Donna's adventures, will start next chapter.
Donna had found him in the console room, fiddling with some loose wires. She leaned over so that from his position under the console, he had no choice but to look at her.
"Hey." she said mildly.
"Hello." he replied in his absent "I'm doing things you don't understand go away" voice.
Donna leaned over a little farther. "Entertain me."
The Doctor paused mid-wire-fusion to raise an eyebrow. "What, running about and saving the universe isn't enough for you?"
Deep breath. "No, not really." She folded her arms over her chest and gave him a stern look.
The Doctor rolled his eyes. "What do you want to do, then?"
"Oh, I don't know. Is there a T.V. somewhere on this thing? I've been missing my game shows. You ever seen Big Brother?"
The look of sudden absolute horror that crossed his face at that point threw Donna off. Of course her crazy Martian boy would have trauma issues from a simple reality show.
"Alright, then, no Big Brother. The Weakest Link, maybe?"
As soon as the words were out of her mouth, the Doctor's entire face fell and his lower lip actually wibbled. Donna gave up on making suggestions.
"Oh, I'm sorry. Come on up here and give me a hug. Come on."
The Doctor obediently pulled himself up and buried his head in Donna's neck for a moment. After a few seconds, during which Donna felt a suspicious bit of moisture on her neck, he pulled back looking as cheerful and unbothered as ever. "Well, entertainment, I can do that. As a matter of fact, I have an entire library tucked away the back of the TARDIS. I'll bet you can find something interesting to look at!" And he was off, pulling her by the hand through the winding corridors of the TARDIS.
It turned out that he did have an entire library, bigger than the one Donna had visited as a child in London and outfitted with comfortable furniture and a fireplace.
The Doctor curled up in an armchair with a collection of Shakespeare's sonnets--"Well, I loved Shakespeare before, but now that I know he wrote half his sonnets about me and Martha it's even more interesting!"--and Donna went in search of a good book.
After several minutes of searching, she found a suitably interesting book and made her way back to the little common area where the Doctor was still buried in Shakespeare. Donna settled on the couch next to his chair with "A Comprehensive Study of the Human Psyche as Compared to the Time Lord's Mind", apparently written by a lopsided concentric circle. Being the kind of person who feels compelled to look at every single page in a book, Donna opened the front cover of the book to search out a publication date. She didn't find one, but what she did find was much more interesting. There was a note written on the inside of the front cover.
"Susan," it began, "I hope that this book will sate your desire for knowledge of humans. Hopefully, once that great thing happens, you will become enamored with a species that lives on a more interesting planet."
Donna snickered. The writer was obviously a Time Lord. Apparently they were all as amusingly pompous as the Doctor. The end of the note was intriguing, though--"All my love, Grandfather".
"Oi," Donna said loudly, poking at the Doctor's knee, "lookit this."
The Doctor obediently dug himself out of his book and peered at the book that Donna was holding open for him.
"Who were they, then, Susan and Grandfather?" she asked him curiously. "I suppose you ran around with Time Lord companions before you came to Earth proper?"
The Doctor snorted derisively. "Oh, not hardly. Susan was my granddaughter. The first time I came to Earth it was because she was fascinated with humans." then, at Donna's disbelieving look, "No, really. Back then I had white hair and everything. This certainly isn't my first body."
Donna grinned, thinking of her own white-haired grandfather. "Did you take her stargazing?"
The Doctor stared into the fire, a sudden far-away look in his eyes. "Stargazing? No. Oh, Donna, I took her to the stars..."
After that he had joined her on the couch and he spent the rest of the evening telling her all about the adventures he had shared with his granddaughter. He turned out to be a brilliant storyteller, filling his tales with sound effects and wild gestures, sometimes even leaping off the couch to act something out.
Hours later, when the Doctor had finally talked himself out and he and Donna were slumped against each other on the couch giddy from laughing until they cried and crying until they had to laugh again, the Doctor pulled himself up, patted Donna on the arm, and shyly admitted that he hadn't had so much fun in a long time before disappearing--not, to Donna's surprise, back to the console room, but to his own bedroom.
It was a turning point in their relationship.
The next time she went looking for him he produced a pack of playing cards and taught her how to play Venusian blackjack, and the time after that she complained that she was hungry so he took her to the kitchen and showed her how to prepare Blue Broccoli Casserole which was a delicacy of the Deigns system, and the time after that they spent four hours playing dress-up in the wardrobe room, and the time after that the Doctor sheepishly explained that he really did need to do some work on the TARDIS but Donna could help if she really wanted, so she ended up learning the names of half the Doctor's tools and by the time they were finished she could've recalibrated a whatsit all by herself.
That night, before they went to bed, Donna bumped the Doctor's shoulder purposefully. "I had a good time tonight, mister."
The Doctor gave her a friendly bump back. "Quite right, too."
And there they were.