On my Note 2 (rooted) I was able to just change a .conf file in one of the directories and the loudness increased by about 300%. It is so loud that it actually hurts my ears when at full volume resting on my leg. There's absolutely no distortion. I can hear music clearly from one side of the house to the other.
So, I suspect that the Note 3 will be the same. Unfortunately, one must root the phone to make a change to the .conf file.