Magento – Fix Mage Registry Key ‘_singleton/magenotification/observer’ Already Exists

I don't have a solution, but I can explain why/when this is happening.

TL;DR: In your case the model admin/session is not instantiated or does not exist and Mage::getModel('admin/session') returns false

Full explanation

There is a bug (or better yet an inconsistency) in the Mage::getSingleton method.
It has been there for ages.

The method looks like this:

public static function getSingleton($modelClass='', array $arguments=array())
{
    $registryKey = '_singleton/'.$modelClass;
    if (!self::registry($registryKey)) {
        self::register($registryKey, self::getModel($modelClass, $arguments));
    }
    return self::registry($registryKey);
} 

Now let's take a look at the Mage::register method

public static function register($key, $value, $graceful = false)
{
    if (isset(self::$_registry[$key])) {
        if ($graceful) {
            return;
        }
        self::throwException('Mage registry key "'.$key.'" already exists');
    }
    self::$_registry[$key] = $value;
}

Notice the throwException. It is exactly what you see as error message. If there is already a value in the registry for a certain key, you get an error.
In your case you get this error because there is already a value for key _singleton/admin/session. This happens in one case.
Going back to the getSingleton method, this is the bug/inconsistency:

 if (!self::registry($registryKey)) {

This condition will return true when the value you registered for a specific key is null, false, 0.
This happens when getModel returns false. (when a model class is not found).

You should check if you have extensions rewriting the admin session class, or you can debug starting from the methods above to see why you are getting this error.

It's most likely caoused by some extension you have installed.

Try to disable the new extension you installed and disable the compiler. Then re-compile again.

You can do this from the admin panel or if it's not accessible try those commands:

php -f shell/compiler.php disable
php -f shell/compiler.php clear
php -f shell/compiler.php compile

You can also diable the module in app\etc\modules directory.