Twist in the proper direction

Are you turning the right way? Remember, "Righty tighty, Lefty loosy" (which never made sense to me). Most screws are right-handed threads, so you'll turn them anti(counter)-clockwise to remove them. In some situations left-hand thread screws are used (usually to prevent screws from loosening due to rotational forces), so you'll actually have to turn the screw clock-wise to loosen it.

Sometimes tightening the screw very, very slightly, can loosen it up enough that it can be removed.

Muscle

The first thing to try, is simply to apply more inward force while turning.

• Use a screwdriver with a head that fits snugly in the screw head.
• Put the palm of one hand on the back side of the screwdriver handle, and wrap your other hand around the handle ready to twist.
• Push the screwdriver as hard as you can with your back hand into the screw, and twist with your other hand.

If you are using a screwdriver with a hexagonal or square shaft, you can grip the shaft of the screwdriver with an adjustable spanner (wrench) or vise grips. This will allow you to exert more torque, and may put you in a better position to push the screwdriver harder into the screw head.

Lube

Sometimes screws seize up due to rust, or other debris. Soaking the area with penetrating oil, may free up the screw enough for you to remove it.

• Apply a liberal amount of penetrating oil to the area.
• Allow the oil to soak in for 5-10 minutes.
• Try to remove the screw (by conventional means, and using the technique above).

Rapping, Tapping, and Banging

If both methods above have not worked, giving the area a sharp tap with a hammer might break the screw free.

• Hit the screw and surrounding area with a series of sharp taps with your hammer.

Remember, you're not trying to knock the screw into next week, you just want to loosen it up (though there are situations where you'll have to give the piece a healthy whack).

Impact Driver

For really stuck screws, you can use a manual impact driver.

• Attach a bit that fits snugly in the screw slot
• Set the driver to spin in the proper direction (this process varies from tool to tool, but most commonly, you'll compress the driver and twist).
• Place the driver on the screw, and hold the driver as straight as possible (keep you hands away from both ends of the tool).
• Give the back side of the driver a whack with a hammer (a rubber mallet or non-marring hammer is good for this).
• Repeat until the screw starts to twist (most impact drivers have to be "reset" between strikes, so don't forget to reset the driver before whacking it again).
• Once the screw is freed up, use a regular screwdriver to remove it.

This may not always be the best tool when working with sensitive electronics or delicate equipment, but it's very useful for removing stuck screws.

Drilling

If you still cannot remove the screw, as a last resort you can drill the screw out.

• Select a drill bit that is just smaller than the shaft of the screw you want to remove (you may have to guess if you can't see the whole screw, but you can always go bigger later so start small).
• Place the bit in a drill.
• Place the tip of the bit in the center of the head of the screw.
• Drill until the screw comes free, or until you think you are deep enough.
• If the screw did not come free, switch to a slightly larger bit and drill again.

This is a last-ditch effort, and will destroy the screw (and possibly the threads in the hole, if not done correctly). Use this technique only as your last resort.

Explosives

Remove the screw, and everything else in a 20ft. radius.

• Place the charge on the screw.
• Light the fuse.
• Run like hell.
• Cover your ears.
• Bask in destructive glory.

It's a good idea to film the explosion using a high-speed camera, so you can watch it over and over again in super slow-mo.

Ratchets can be rather bulky as you've found. Combined with a deep socket in the wrong place, you aren't going to remove the fastener. Using a shallow socket and and proper length extension moves the ratchet out of the way to where it belongs.

Straight Drive Extension

Wobble Drive Extension

The wobble drive allows for some misalignment between what the socket is removing and the ratchet drive handle, probably what you're looking for to take care of the fact that gear is in the way.

Or you use this drive handle to convert the socket to a simple nut driver

Then there's the time honored nut driver