Take a look at Microsoft's KB on the issue:
When you start a computer that is
running Windows Vista or Windows 7 for
the first time, one or more of the
following icons do not appear in the
notification area in the lower-right
corner of the screen.
- The Network icon
- The Volume icon
- The Power icon
You may also notice that the check boxes for the related system icons are
not selected on the Notification Area
tab of the Taskbar and Start Menu
Properties dialog box. However, the
network, volume, and power features
function correctly even though their
associated icons do not appear in the
notification area. You can access
these features by using Control Panel.
If you gave your computer enough time
to finish the startup service
initializations before you restarted
it, and one or more of the system
icons still do not appear in the
notification area after the restart,
you can make them reappear by deleting
two subkeys from the registry.
Microsoft provides a "Fix It For Me" file that will automatically make the changes needed to fix this problem. You can download the Fix It For Me solution from here, or follow these steps to fix it yourself:
Click Start, type regedit in the Start Search box, and then press ENTER. If you are prompted for an administrator password or for confirmation, type the password, or click Allow.
Locate and then click the following registry subkey:
HKEY_CURRENT_USER\Software\Classes\Local Settings\Software\Microsoft\Windows\CurrentVersion\TrayNotify
In the Details pane, click the IconStreams registry entry.
On the Edit menu, click Delete, and then click Yes.
In the Details pane, click the PastIconsStream registry entry.
On the Edit menu, click Delete, and then click Yes.
Exit Registry Editor.
Either restart the computer or the Explorer.exe process.*
*The easiest way to restart explorer is to click the start menu, press Ctrl+Shift and right-click on the empty area of the start menu and select "Exit Explorer".
Then press Ctrl+Shift+Esc, click File -> New Tasks (Run) and type explorer and click OK. Close the Task Manager when you're done.
Make sure the Always show all icons and notifications on the taskbar is unchecked at the bottom of the Notification Area Icon panel. If it is checked, it will grey everything out.
If the above didn't fix it then check under the Start Menu and Taskbar policy path for Hide the notification area, Prevent changes to Taskbar and Start Menu Settings, and Lock all taskbar settings policy settings. Play with this settings. Remember to set it to not configured if these settings fix the issue. This will prevent future issues related to group policies.
Also, your user could a member of another group, besides the administrators group, that might have same policy *enabled**. If that isn't then maybe there is a 3rd party application that is installed and prevent you from change it. There is an example of this at this post.
Information About the Group Policies Refer to Above
More about the Hide the notification area group policy...
This setting affects the notification area (previously called the "system tray") on the taskbar.
Description: The notification area is located at the far right end of the task bar and includes the icons for current notifications and the system clock.
If this setting is enabled, the users entire notification area, including the notification icons, is hidden. The taskbar displays only the Start button, taskbar buttons, custom toolbars (if any), and the system clock.
If this setting is disabled or is not configured, the notification area is shown in the user's taskbar.
Note: Enabling this setting overrides the "Turn off notification area cleanup" setting, because if the notification area is hidden, there is no need to clean up the icons.
More about the Prevent changes to Taskbar and Start Menu Settings group policy...
Removes the Taskbar and Start Menu item from Settings on the Start menu. This setting also prevents the user from opening the Taskbar Properties dialog box.
If the user right-clicks the taskbar and then clicks Properties, a message appears explaining that a setting prevents the action.
More about the Lock all taskbar settings group policy...
Prevents the user from making any changes to the taskbar settings through the Taskbar Properties dialog.
If you enable this setting the user cannot access the taskbar control panel. The user is also unable to resize, move or rearrange toolbars on their taskbar.
If you disable or do not configure this setting the user will be able to set any taskbar setting that is not disallowed by another policy setting.
Best Answer
As mentioned elsewhere, this is not how the Notification area is designed to operate. Notifications are its raison d'être, so being able to turn them off would be counter to the design goals of the system.
What it seems you really want to do is use it to open the program and don't care about notifications. For an equivalent solution, I would try something like this:
This should work. I don't know the particular program you are struggling with, but those that I use often will not launch a new instance using this method, but open the existing instance. I believe that the program would have to instruct Windows specifically to not do this, though what mechanism that utilizes and whether this is true at all, is something I am not intimately familiar with.