Step One: Are you SURE it's just turning and turning...

SORRY! I have to ask! You'd be shocked how much they come out just by a little gentle encouragement.

As a basic first step and periodically throughout, try shaking, pulling or prying the screw out with your knife, prying tool, etc. This might cause a weakened-enough screw to come out. Try sticking a little Scotch tape over the head and pulling during this stage. No? Proceed to next step.

Okay, if you're here, that ain't work. OKAY! That's fine! Try the next step: Forceful realignment.

Get your utility knife, spudger, flathead screwdriver, whatever's flat and strong and get it under the head, and manipulate it around a variety of directions while unscrewing. This can cause it to get removed and I assume it's because of a loose and weakened thread.

If that doesn't work, or if it's too low to get under, proceed to next step.

Not that?! Okay, so, now we're getting to the slightly more destructive options.

Do you feel it wigging, getting loose but not quite, anything like that? Okay, get your craft stick (or whatever you chose). Deposit your cotton/absorbent material or Scotch tape around the screw to avoid accidentally gluing your screw into the socket (and thereby worsening your situation, BADLY...)

Put a SMALL AMOUNT of the strong adhesive on the head of the screw and wait for it to dry. Periodically check up on it in case it's leaking over the side. Then, once it's bound (might take a good long while), yank it out. This might get the screw to just rip out caveman style. It's not elegant but it might work.

If you have one, this is a good time to opt for a Dremel screw extractor. Cut off the head, cut into the hole, whatever it takes to get it out.

I don't know much about dremels but it's going to be very wise to not get too trigger happy as there might be an important component (or worse, a live wire) under the screw.

At this point, you must be SUFFERING. I'll drop the silly quirkiness... you have an extremely bad screw. It might be wise, if you don't really care about the laptop and just want, say, the drive or RAM out to just remove all the other screws and bend the bottom panel to allow access.

Or, just score and break it if it's just plastic. Nobody's looking.

If you really can't damage the thing and all of the previous steps don't work, taking it to an actual repairman will probably be the best option. Plus, if they tell you what it was, you can leave an angry comment here and I can edit it into the guide!