Ubuntu 13.04 Pidgin Tray
Solution 1:
I found that if you enable the Libnotify Popups plugin (pidgin-libnotify) in Tools->Plugins then this problem is resolved. You can set your status via the system tray menu and you can also close the Pidgin Buddy List window and it will remain running. If you don't want notifications you can also configure this plugin in the Plugins menu.