Not only are egg cartons no good for sound proofing but they are a big fire hazard.

First priniples of niose control is to prevent the noise from happening in the first place, not an option in this case. Second principle, remove the source of the niose, not an option in terms of location, but possibely in terms of time. Can you find a time to listen to your music when it doesn't bother your neighbour?
Well that's the free options..now to the more costly ones....
You need to consider how the sound is getting through to your neighbours apartment. Obviously through the wall, but also through the floor and other walls. the sound can travel through any common beams etc. that you share with your neighbour (these are called flanking paths), if you haven't already I would suggest isolating you speakers form the floor. this would help prevent the sound traveling through the floor beams ( and should improve the sound of your system).
Other flanking paths can occur if you have both have windows near each other. It would pay to keep these closed.
There are two main things to think about when you want to have a soundproof wall, mass (the heavier the better) and completeness (no holes, ideally it should be air tight). Check to see if there are any gaps for air to get through and fill them up. You could also try adding mass to the wall by attaching another piece of plasterboard to your side of the wall.
Check out 'The master handbook of acoustics' by F. alton everest.
Also, Noise is a very subjective thing, if your neighbour knows that you are taking him/her seriously and trying to find a solution they may not find the music/noise so annoying.
good luck
-Iain