I've got a stack of 4 flat, white doors. Anyone have any cool project ideas for them?

You could use them to make a backyard shed, perhaps

Artist's shelves... use 2 - 2x4's the same length as the door is wide to space between doors. Put these spacers about 18" from the ends of the doors. Stack as many doors with spacers as you want (and still be stable). Voila! Deep shelves for paper, prints etc. When you move, just unstack them.

Ahhhh, you have the perfect items for an incredible project. I have used those flat panel doors for the most attractive and useful things. It's not the project that you need to think about but what you might desire as a completed item because the doors are made up of material that has so many uses. Also consider the addition of more junk, such as throw away paint, anything from the local dump or dumpsters behind the downtown stores. I often would make a run into town and bring back an absolute load of discards. Just anything that seemed sorta useful. Then, by cutting the edges off the doors and making that into small framing, the leftover sheet of doorskin can be used with the small framing to make anything you can imagine, and there are tons of these doors available to the person that scouts them out. I have made a nice spice rack, some highly unique shelves, a run from floor to ceiling for our cats, several beautiful 3D wall sculptures, perhaps a dozen large lamps. For the lamps, cut the doors apart and design a series of hexagons as large as you can make from the door panels. Each door will yield 4 panels. Cut the all the same shape and cut a big circle from the middle. Put them all together to form a globe of sorts and cut a five inch circle out of the bottom so you can get inside. Use the door framing to make 1/2x1/2 framing to make it stronger inside, then paint it all flat black. Add a light inside and cover all the big holes with plastic on the inside. Take the circles you cut out and the 1/2x1/2 frames at about two inches and make the removed circle cover the circle under it. Cut out the center of that one and sit it 1" above the other, then do another progressive step but leave the center missing. Make this into a lamp and hang it from the ceiling. They really look great! t I also did a large greenhouse with removable panels using about 20 old flat doors. I cut out the center and filled in the gap with parts of other doors, then routed a small rabbet for the acrylic to drop into. The doors had about a six inch edge all around, but otherwise were all clear, and each door would come off the frame, or raise depending on how I had it setup. I made the opening sort of oval, so it looked nice, painted them green, and set them on 2x6 frames. Thew sides were greenhouse plastic and the front door matched the roof panels. Worked great.

I seem to recall that there are a few instructables of people making computer desks out of doors. Being hollow doors are not terribly solid...

They usually come in handy as scrap material for temporary jigs and such for me. A hollow door isn't much use for anything but as a door most of the time... however I did see a rather nice outdoor planter constructed out of four doors (two upright and two horizontal forming a box with two higher sides). They were solid, but your doors could probably be reinforced. If they're all the same size, I'd make a folding smoke-house (smoker). Find a nice flat area and hinge them all together so that they can be folded together into a stack or opened up into a box-shape (may require a pair of 2 way hinges). Add a few rails on the inside of the box to hold wire racks. Use as a smoker for making jerky or smoking fish/game/etc. Fold it flat for storage in the off-season.

A door set atop two stacks of cardboard copier-paper boxes can form a useful, attractive, and surprisingly strong desk (and it has storage too).

a dining room or coffee table? or as you said, paint on them - you could do a 4 panel mural for your basement