I need help figuring out the most efficient way (NOT electronically) to manually compress plastic in a small confined space. I was thinking there were small compression shocks that would help to triple my human body force but have been unsuccessful in finding something that works.
Levers are your friends: a lever allows you to multiply up the force of your own strength. Here's a manual can crusher, demonstrating this:
Your hand moves three or four times the distance that the crushing part does; as a result, the crushing part moves with three or four times the force that your hand does.
A bench press or arbor press, combined with a suitable die for horizontal containment, will allow you to produce rather high forces and pressures.
Do you have any numbers — volume, pressure, etc. — regarding exactly what it is you're trying to achieve?
EDIT: I have to confess that I missed the waste-disposal tag on my first reading, so my first suggestion isn't very practical in that application. Probably the most compact mechanism that can provide both the required force (about a ton) and the travel distance (18" to 24") is the "scissors jack" — but it requires a lot of cranking to operate that might not sit well with users.
I'm not clear from your question if you are trying to crush a plastic object, or manipulate hot plastic (eg for injection molding.) Lever systems are certainly a good option as has been mentioned in other posts, but depending on your application, a compact lever may not provide enough force for you..
If you need more force, hand-pumped hydraulic jacks will let you get a serious amount of force in a small space. For example, Enerpac makes a 2.25"x1.5" cylinder that generates 5 tons of force or a 5"x5" cylinder that generates 60 tons of force. Hydraulic hand pumps are readily available although the cylinder will move very slowly.