Method: Remove spring-diaphragm amplification
Target audience: This is especially useful for standardized test takers (MCAT, SAT, LSAT, GRE, etc.) and others who need the element of surprise in whatever they are timing.
Step 1: Get a cheap watch somehow
Perhaps you can find a digital watch around the house.
Step 2: Remove rear plate
You will likely need a phillips pattern electronic (watch size) screw driver for this. Flat head could also be improvised to work very easily. I highly recommend the Stanley 6-pack of screwdrivers with rotating top for convenience.
Step 3: Identify and remove small amplifier spring
Find and remove that spring and the beeping will stop. If you are lucky, the spring will simply fall out when you open the watch.
You could also remove the piezo that is glued to the inside of the cover but as you can see from my scratches, that is more difficult.
Step 4: Reassemble
Test the buttons to ensure the beeping has stopped.