It seems that Android DSP apps all do different things and you'll have to try different ones to see if they work for your particular phone and ROM. I haven't had any luck tracking down why this is the case, or even what the apps are doing. One time I even tried modifying a config file that by all appearances controlled volume levels, to no avail.
I have found one that works for my Note II running OctOS L, and it's called Volume Booster. It's ad supported, and occasionally stops working, but it's great. I boost about 25 - 30%, which is loud enough to hear music across a room at a decent level. At that level, there are sometimes some distortions. Also, I think it negatively impacts phone audio, but since I don't make many calls, I haven't looked into this problem.
Given the distortion, I think that this app meets your requirement to get louder than the manufacturer intended. I presume that if I boost too much with this app, it will blow the speaker.
From the Play store:
Simple, small, free app to boost your speaker or headphone sound
volume. Useful for movies, audio books and music.
Use at your own risk. Playing audio at high volumes, especially for a
prolonged amount of time, can destroy speakers and/or damage hearing.
Some users HAVE reported destroyed speakers and earphones. If you hear
distorted audio, lower the volume (but it may be too late).
By installing this application you agree that you will not hold its
developer responsible for any damage to hardware or hearing, and you
are using it at your own risk. Consider this to be experimental
software.
Not all devices support this software. Try it at your own risk and see
if yours works.
This app does not work on most 4.2.1-4.3 devices. It should work on
4.4 and higher, as well as on devices below 4.2.1.
This is not for adjusting the speakerphone volume in phone calls (that
has its own boost, I think), but for adjusting the volume of music,
movies and apps.
When you set the boost to zero, Volume Booster will be off. The
notification icon is just for ease of launching. If you don't like
seeing the notification icon when Volume Booster is off, just go to
Volume Booster's Settings and set it to appear only when Volume
Booster is running.
