This door does not like to latch; to latch it I must pull sharply up on the doorknob. So I see a shim in my future. I suppose I need to unscrew the bottom hinge, but what sort of screw-driver does that strange square hole? (Alternatively, would I just shim on the door side?)
I agree it is a stripped Phillips screw head, and that the head should be drilled off, the hinge removed, and then the remaining screw shaft unscrewed with a pair of pliers.
"Sometimes", you can get or grind a blade screwdriver whose tip is exactly as wide as the Phillips cross. Remove all the other screws, insert the blade tip into two of the remaining tips of the cross, push on the back of the hinge to get some extraction force going, and turn the screwdriver until some of the screw head is protruding. Continue to work the screw out this way, or alternately grab the protruding head with pliers. Another trick is to remove all the other screws, and, with gently increasing force, pry the back of the hinge until the wood gives way and the screw comes out. Then repair the hole with white glue and a dowel, or a few chips of wood, or even a plastic expansion anchor or molly bolt
sears, has a driver set I have used several times it was 12$ for 3 sizes here is a link hope that is allowed: http://www.craftsman.com/craftsman-3-pc-screw-out-174-damaged-screw/p-00952154000P?sid=IDxCMDFx20140801x001&KPID=00952154000 there have been a few times I had to drill out the head then use vice grips to grab the shaft and unscrew that way where I did not want to move a hinge. Another cool thing to have for screw removal is a small set of lefthanded drill bits, some times when drilling the head they bite in and back the screw out, but they are harder to find and more expensive than the set I got from sears
As others have said, it's a stripped Phillips Screw.
What's not quite so clear is why it's there. My guess is that someone fitted a brass screw (as opposed to the others which do seem to be steel, at any rate their heads are smaller) in that position, with a power-screwdriver. Then, realising it was proud and stopped the door closing, they tried to remove it, but being brass and soft it stripped out, so they left it.