Seem like no one has a solution yet, although many are experiencing the problem Stardock really has to fix this. They shouldn't allow this become a reason to not adopt an otherwise excellent product.
If Start8 has been removed from the machine (please check all the files in the Start8 folder are removed (if the folder even exists still) then it is doing nothing on the machine.
Could you possibly have installed another product (for example Stardock Decor8, Stardock ModernMix) at the same time and the issue is somehow linked to those? Please check to see if they are listed in the installed programs you can uninstall.
Where else have you seen many other people with this issue? I see two people in this thread and it seems the two of you are not reporting the same thing either. One is after the expiry of 1.16 trial, the other is it all worked fine until upgrading from the trial to the full version. I am aware of two other threads which were bumped by allnycyachts too but again one of them is totally different and the menu opens fine, it just doesn't open any Win8 apps.
Given the number of trials started since Start8 1.16 was released you would anticipate many orders of magnitude more reports if Start8 was directly responsible. If you then do a google search for "Windows 8 unable to open apps" you get many results and Start8 does not seem to be in use.
For example : http://www.eightforums.com/software-apps/21010-windows-8-apps-wont-open.html
I suspect the problem is the OS corrupts the user account at some point and because you are in a trial of Start8 you don't notice the issue until Start8 expires and so you are forced to go back to using the Windows 8 start menu. Additionally for those who encountered the problem after installing an update, it is probably a case of rebooting the machine showed up something that wasn't seen before as Windows 8 usually uses its hybrid shutdown and so doesn't actually fully reboot and start clean. It only starts clean after a full reboot as would happen after software installation or software uninstallation.
One interesting thing to try would be to create a new user account and then log into that and see if it works. If so it is most likely a corrupt user profile which Windows 8 seems especially prone to.
Please do contact support@stardock.com so more data can be collected as this would allow us to confirm that our software did uninstall correctly and help rule out any possible conflicts between software such as AV apps.