Almost 99% of the time there is no need for an Android application to take over its own life cycle. Most of the time it comes down to better planning or smarter design of the application. For example, rather build an internal service (not exported) to handle downloads, etc., or design actions and tasks around user workflow.
But that being said, where there is a will there is a way. Android provides - through the android.os.Process class, a much better API than Java to control the underlying process. And unlike Java it does not treat the developer like a moron by hiding it all behind a simple java.lang.System.exit() call.
So how do you ask your application to commit suicide in Android? Well, the trick is simple:
Create your own Android application class by inheriting from the standard android.app.Application class (remember to declare it in the AndroidManifest.xml file).
Override the onCreate() method, and store the process ID which started your application:
this.pid = android.os.Process.myPid(); // Save for later use.
Now to kill your application, provide a kill() method:
android.os.Process.sendSignal(pid, android.os.Process.SIGNAL_KILL);
Now whenever you need your app to commit suicide just type cast the application context, and call your kill method!
((MySuicidalApp) context.getApplicationContext()).kill()
Just remember that due to the process management policies in Android, specifically related to services, Android may just opt to restart your service (see You should not use task killers on Android).