Do a backup, using either Time Machine or a cloning program, to ensure files/data can be recovered. Two backups are better than one.
Try a restart.
Run Disk Utility/Repair Permissions. Run until it reports no errors. If there is a recurring error, see the article on what permission errors can be ignored.
Try setting up another admin user account to see if the same problem continues. If Back-to-My Mac is selected in System Preferences, the Guest account will not work. The intent is to see if it is specific to one account or a system wide problem. This account can be deleted later.
If the problem is still there, try booting into the Safe Mode. Shut down the computer and then power it back up. Immediately after hearing the startup chime, hold down the shift key and continue to hold it until the gray Apple icon and a progress bar appear. The boot up is significantly slower than normal. This will reset some caches, forces a directory check, and disables all startup and login items, among other things. If the system operates normally, there may be 3rd party applications which are causing a problem. Try deleting/disabling the third party applications after a restart. For each disable/delete, you will need to restart if you don't do them all at once.
You have some 3rd party application that has been reinstalled after the restore.
Start by removing everything from here:
> System Preferences > Users & Groups | Login Items
Select all the items in the Login Items list,then hit the minus '-' button underneath the list (do NOT check the 'Hide' box).
Next, restart your computer.
Affer the computer boots up, test to see if the problem remains.
Something really stange going on here. My disk test show drtive is in great shap, I have removed all login items. System boots just fine and all native apps seem to load and run. I installed a fresh install of Chrome. When Chorme opens it only stays for about two seconds then shuts down. It seems to be failing exactly the same way Skype did in previousn rounds of testing. Are you or anyone you know using 10.8.3 and successfully running Chrome, Skype and other 3rd party apps? I concerned if I go a buy a new Mac with 10.8.3 installed I will come home with the same exact issues.