I've been wanting to write something like this for ages.
By the way, the reason the 'Ah Elbereth Gilthoniel' poem is in English is, obviously, the Tardis translation matrix :)
(Edited to enhance readability)
The Doctor sat by the River of Nimrodel, listening to its sweet music. It was very relaxing.
"Ah, Elbereth," he sighed. He'd read the books, obviously.
"O Elbereth Starkindler," a voice at his side began.
"White-glittering, slanting down sparkling like a jewel,
the glory of the starry host!
Having gazed far away
from the tree-woven lands of Middle-Earth,
to thee, Everwhite, I will sing,
on this side of the Sea, here on this side of the Ocean!"
"Hang on," he said, looking at the elf lady beside him. She had appeared so silently he had not heard her approach. Which was very rare for him. "Who are you?"
"I am Arwen," she said with a gentle smile. "And you are the physician of the Universe, the Doctor you like to be called."
She was beautiful. In all his years of travel, he had never seen anyone so lovely, so fair, so utterly exquisite that he could not even find words to express it. This was really, really rare.
"Oh fair maiden," he began, in a besotted voice. It was probably something like the way she would expect people to talk in this time and place. "How lovely thou art, as if Eru himself had formed thee of sunshine and silver and not of the common elements. Thy raven hair is passing fair, thy eyes they shine like moon on wine, thou art white as a lily and tall as a ..." here he stopped. Partly because he couldn't come up with a good simile, partly because she was staring elf-daggers at him. Nasty things, elf-daggers.
"I beg pardon, lady, if I have offended."
"Good grief, Doctor," she said. "What in the world kind of speech is that?"
"Erm ... How about this? Baby, you are my everything. Don't go away, I need you to stay, I need you every day. Baby, I love you."
"I'm really not a baby," she said, with a spark in her eye as of a mother listening in amusement to a little child babbling on. "I'm hundreds of years old."
"Well, so am I," said the Doctor. "I'll bet I'm older than you."
"Perhaps you are. Yet your life is also fuller of grief than mine."
"I had forgotten all my sorrows in the depths of your dark eyes."
She rolled her dark eyes. "You have no claim on me. I am already betrothed."
"Oh, pshaw. Aragorn? All he has to offer is a kingdom. I can show you all of time and space; the history and future of this world and all worlds; the singing towers and the swaying mountains, the Judoon and the Daleks (well, not the Daleks perhaps), Raxacoricofallapatorius and the Library. Everything and everywhere and everywhen. Come with me, Arwen Undomiel, fairest of all who walk or have walked or will walk this universe. My life was dark until you came to me, meaningless, purposeless. I was like a ship adrift in the ocean of time and space ... "
"Well, perhaps you should run that by Aragorn first," said Arwen. "I don't think he'd be too happy to hear that."
"He couldn't do anything about it. He is primitive, I have a space ship that dematerializes in seconds."
"I think you'd find he would invent time and space travel to get me back. Plus, I like it here. Your eyes show the weariness of your journeys."
"But I have to have you, oh maiden of beauty! You do not grow old, you do not die. Aragorn will one day die, and you will feel the pain that I have felt far too often. I am ready for someone who will stay with me forever."
"Your protestations of love are sweet," she said. "But I am of the elven kind (well, three fourths elf). We do not die, nor do we change. Yet you will die, Doctor. And you change. You change many times, Doctor. How could I bear that?"
"Oh, Elves," he grunted. "So set in their ways. Can't abide a little change now and then."
"Oh, Doctor, running forever," she replied. "You can never sit still for a moment and be at peace."
"Well, I was doing so until you so rudely interrupted me."
"Oh, so now I'm rude am I?"
"Your face belies your character, elf-kid."
"Aragorn!" Arwen shouted suddenly. "Come and meet my new friend!"
Aragorn came towards them. He took one look at the Doctor and drew his sword.
"Get thee away from Arwen, Doctor-of-many-faces! I know that thou wishest to draw her away with fair words and cute smiles, but I will not allow such poltroonery."
"You don't mind when Aragorn talks like that," said the Doctor, backing away from the sword.
"That's because he's my honey," said Arwen. "Plus, he has a much more pleasing voice than yours."
"Oh, really? Aragorn, I dare you to a sing-off."
"That's easy," said Aragorn. "I won the Tri-Kingdom Vocal Contest three years in a row."
"What shall we sing, pop or opera?"
"Opera, of course," said Arwen with a sly smile.
Aragorn gave her a loving (albeit cheesy) glance.
"As the challenged, I will allow you to sing first," he said to the Doctor.
The Doctor cleared his throat; now that it came to it, he was none too sure that his voice could stand up to the tough old king of Gondor. Then he launched into O Sole Mio. He wished he had some music to accompany him, but he thought his singing was rather good. As he progressed further into the song, he grew more confident, and he fairly belted out the last verse.
At the end he bowed deeply. "Beat that one, bearded fellow."
He was surprised when, instead of singing, Aragorn doubled over with laughter. Arwen followed, laughing so hard that she almost tumbled into the stream. The Doctor watched them carefully, wondering if perhaps they had consumed alchoholic beverages before meeting him. Then he became alarmed when they showed no signs of stopping. Aragorn's laugh was deep and throaty, while Arwen's was like ... well, he would have said like silvery bells if he had still been besotted. As it was, he said,
"You sound like a cow's bell, Arwen. Shut up, the both of you!"
They both shut up and stared at him. Arwen had tears on her face. Aragorn quickly turned away to wipe his off; tears, even of laughter, were unmanly in his scroll.
Arwen breathed a huge sigh. "I haven't laughed like that in ages," she said. "That was hilarious. I could hardly breath. Wasn't that funny, Aragorn?"
"Yes, my love. My sides hurt, and my breath seems short."
The Doctor was growing more and more angry; his wounded pride rose up.
"I'm a Time Lord, I'll have you know! I've seen and done more things in my lifetime than your pathetic minds could ever dream of!"
"Orcs," said Aragorn.
"Slitheen!" said the Doctor.
"A balrog of Morgoth!"
"Shadows that eat you!"
"Ringwraiths!"
"I towed the Earth home!"
"I confronted Sauron in the Palantir!"
"I confronted a bunch of eye-ball Atraxi, scolded them, and sent them home!"
"I led my people to victory at the Black Gates!"
"I destroyed MY people!" Oops, that one wasn't supposed to slip out.
Arwen drew back in horror, while Aragorn brought his sword back up level with the Doctor's throat.
"I destroyed the Daleks, too!" It was better than nothing.
"I resisted the lure of the One Ring!"
"I resisted the lure to create massive paradoxes to get the things I wanted!"
"I captured the slippery creature Gollum!"
"I stopped the Master from destroying the Earth and the whole universe with the Toclafane!"
Aragorn apparently couldn't think of anything to say to that, so he stepped back.
"I own you to be the more adventurous man," he said, quietly. "Yet," in a louder voice. "Methinks you have not such a jewel as I have here, for all your travels!"
"Ha!" said the Doctor. "I have the Tardis. She's infinite, she exists in all of space and time, she travels through space and time, we stole each other and we protect each other. Top that one, King of Gondor."
Aragorn truly looked defeated. "Unfortunately, Arwen," he said. "I cannot defend you against such massive proportions as that."
"Are you calling my Tardis fat?"
"Are you calling my betrothed less attractive than a machine?"
"Are you saying that a pinch-faced elf-girl is more amazing than a time machine?"
"On guard, churl!" Aragorn brandished his sword.
"I'll get my sword," the Doctor said. "But I'll have you know, before you get your head cut off, that I defeated a Sycorax leader in single combat."
"I have slain many orcs and men in battle." Aragorn stuck out his chin. "I'm not afraid of a pacifistic physician."
The Doctor went inside the Tardis and closed the door for a moment. Then he stuck his head out.
"So long Aragorn and Arwen! I don't have time for a ridiculous sword fight. Places to go, people to see, ski lifts to ... ski lift on."
"Coward!" shouted Aragorn and Arwen in unison. But the Doctor was gone.
FINIS