AndroidManifest.xml

<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
    package="org.example.response22166282"
    android:versionCode="1"
    android:versionName="1.0" >

    <uses-sdk
        android:minSdkVersion="14"
        android:targetSdkVersion="19" />

    <application
        android:label="@string/app_name"
        android:theme="@style/AppTheme" >
        <activity android:name="MainActivity" >
            <intent-filter>
                <action android:name="android.intent.action.MAIN" />

                <category android:name="android.intent.category.LAUNCHER" />
            </intent-filter>
        </activity>
        <activity android:name="Exiter" android:theme="@android:style/Theme.NoDisplay"/>
    </application>

</manifest>

Exiter.java

package org.example.response22166282;

import android.app.Activity;
import android.os.Bundle;

public class Exiter extends Activity {

	@Override
	protected void onCreate(Bundle savedInstanceState) {
		super.onCreate(savedInstanceState);
		finish();

		// Android will clean up the process automatically. But hey, if you
		// need to kill the process yourself, don't let me stop you.
		System.exit(0);
	}

}

MainActivity.java

package org.example.response22166282;

import android.app.Activity;
import android.content.Intent;
import android.view.Menu;
import android.view.MenuItem;

public class MainActivity extends Activity {

	@Override
	public boolean onCreateOptionsMenu(Menu menu) {
		getMenuInflater().inflate(R.menu.main, menu);
		return true;
	}

	@Override
	public boolean onOptionsItemSelected(MenuItem item) {
		int id = item.getItemId();
		if (id == R.id.action_finish) {
			finishAndHide();
			return true;
		}
		return super.onOptionsItemSelected(item);
	}

	private void finishAndHide() {
		// We don't call finish() explicitly. FLAG_ACTIVITY_CLEAR_TASK should
		// take care of it.

		// Replace the current task with one that is excluded from the recent
		// apps and that will finish itself immediately. It's critical that this
		// activity get launched in the task that you want to hide.
		final Intent relaunch = new Intent(this, Exiter.class)
				.addFlags(Intent.FLAG_ACTIVITY_NEW_TASK // CLEAR_TASK requires this
						| Intent.FLAG_ACTIVITY_CLEAR_TASK // finish everything else in the task
						| Intent.FLAG_ACTIVITY_EXCLUDE_FROM_RECENTS); // hide (remove, in this case) task from recents
		startActivity(relaunch);
	}

}

This worked, although I still had to call finish on MainActivity, after startActivity.