Something that you will eventually get and continue to get once you become a Marine. Getting the "big green
weenie" may include such
fine events as: getting fucked with duty on a particularly important weekend, getting in trouble for some small
petty shit while being in garrison, getting stationed at a shithole like
29 Dumps.
The "big" represents the big bad Marine Corps picking on the little guy, the Marine. The "green" represents the true color of the Corps. The "weenie" represents getting fucked over.