Disclaimer: Don't own Doctor Who. Ordering my own version of DT's sonic screwdriver, though… ;-)
I'll just give another kind of disclaimer right from the beginning. I'm not the funniest person in the world, but everything on this list made me giggle while thinking it up, so I hope at the very least you get a chuckle or two. If the fancy strikes me again, or I think up new/better jokes, I'll post them in another chapter. Also, if there's anything you can think of, send it along in a review or a PM, and I'll make sure it gets credited to you. (I did the same thing with Stargate SG:1, so no worries.)
Things a Time Lord (a.k.a. the 10th Doctor) does for fun:
Instead of introducing himself as the Doctor, say, "I snogged Madame de Pompadour!"
Play hide and seek with the Tardis.
Pay a random stranger to go up to his travelling companion and say, "So… there was
this problem, and I've had to regenerate…"
Same as above, but have the random stranger be a woman!
Pretend he's been locked out of the Tardis and tell his companion they're stuck.
Using his sonic screwdriver, rig everyone's ipod to play "Eye of the Tiger" at the same time.
Land the Tardis in the middle of places that will freak out his companion when she opens the door (solid rock, center of a planet, etc).
Carry around a basket of bananas and toss them to random people.
Order a medic alert bracelet that says he's allergic to aspirin.
Walk around at Christmas with a sign that says, "It's Christmas everyone! Do you know where the Doctor is?"
Grafitti the Torchwood logo so it says, "Thick as a stick of Torchwood".
Land in Buckingham Palace and tell the royal family they're all werewolves!
Buy a pocket watch and give it to Martha saying, "Don't know why this is important, but some doctor told me to give it to you."
Order a custom pair of Converse.
Rig Jack's wristwatch so that if he ever time travels he only shows up in bathrooms.
Send Jack fake secret admirer letters from the Queen.
Dance around inside the Tardis in a white button-down shirt and socks.
Run through a field, kicking all the scarecrows.
Run headfirst into trouble!
Try to land the Tardis on top of a pyramid.
Dress up in the outfits his other incarnations used to wear.
My personal favorites are:
Play peek-a-boo with the heart of the Tardis.
Reprogram a Dalek to say, "I am a talking rubbish bin, and I will exterminate you with my magic plunger!"