Pretend you've never met, then loudly try out lame pickup lines in a swanky bar. Act like they worked.
Go on a walking journey and every fifteen feet draw a chalk arrow in the direction you're going. At the end of the trip, leave a big pile of chalk.
Create photo evidence suggesting that you went on an adventure that didn't really happen.
Go for a drive. You can only make right-hand turns. When you finally get stuck, turn around and then you can only make left-hand turns. Repeat until you find something interesting. Take pictures along the way!
Build forts out of furniture and blankets, and wage war with paper airplanes.
Go to a major chain bookstore, and leave notes to future readers in copies of your favorite books
Write a piece of fiction together. Outside at a cafe. Ask strangers when you get stuck.
Try and visit as many people as you can in one night, and turn as many things inside their apartment upside down as you can, without them noticing.
Do the lamest tourist thing in your area that you have both secretly wanted to do forever. Have an unabashed good time!
Hide and seek in the park
Go around the city with sidewalk chalk and draw hearts with equations inside on random things
Drive somewhere unknown and have dinner in a city you've never been to. With fake names.
Go for a drive with the passenger blindfolded, choosing directions at random. see where you end up
Dress up as pirates, commandeer shopping carts, and have a war upon the high seas.. er, parking lot.
Go on a search for as many good climbing trees as possible, climb as high as you both can in all of them, compile photo evidence
Rent a movie you've never seen before. Set on mute and improvise dialogue.
Dress up as pirates and go parrot shopping at local pet stores
Go to the airport, get the cheapest, soonest departing flight to anywhere when you show up, and stay there for a weekend.
Walk around a city and perform short silent plays in front of security cameras
In the middle of the night, drive to the beach, so you arrive just as the sun is rising. Have a breakfast picnic, then fall asleep together. Bring a sun umbrella.
Dress up as superheros and stop at least one petty crime "ie. jaywalking, littering...."
Go to a minor league baseball game under the stars. Tell each other stories about how bad you are at athletics. Randomly cheer for both teams. Eat lots of Cracker Jack.
With camera and pair of boots, make photolog of a day in the life of the invisible man.
Walk around the city all night and find a place to eat breakfast at dawn
Go to a restaurant and convince the cook to create something completely new for you.