I'm having this problem too.
Ever since a new update has been available, which I install. The program shuts down after install and when I open it up it tells me that there is an update to install.
Tried rebooting, then when I tried delete / re-install it said "Uninstall completed with errors" and never deletes. I'm stuck in a time loop.
I can't download CC apps from the site either as a similar issue occurs.
I am having the same problem as well and have went through the same steps as trev-ireland. Windows 8 user.
I just hit the same thing today. I get error 600 when trying to open Creative Cloud from the applications folder. If I try to uninstall I get 'Unable to shutdown Creative Cloud desktop. Quite Creative Cloud desktop and try again'. I am running a MacBook Pro mid-2009 with Mavericks - 10.9.4.
I got this fixed last night!
I can't find the chat I had with the tech agent last night but we manually renamed and deleted some folders, plus we also used adobecleaner to fully delete the CC app (this is done through an old dos menu)
I'll try find the folder names and method we used to post it here asap.
I'm on windows 7.