I run Windows 7, recently had this problem after the patch like many others, and took two steps to fixing it.
1) There seemed to be an issue with the ActivationUI getting stuck. I tried setting it to run as Administrator and it worked for one launch but not again.
2) I found a number of redundant entries for Dragon Age Inquisition in the windows firewall. I generally don't run a software firewall at all and am unsure why the Windows one was on but I turned it off and have had no problems running DA:I ever since.
I can't say if the admin flag on ActivationUI (the one in the Dragon Age folder) made a difference but everything's worked just fine since I turned off the firewall. I removed the Dragon Age entries from the registry for cleanup but I don't think that is necessary and mucking around in the registry can easily cause more harm than good.
If dropping the software firewall fixes it for anyone else and there are security concerns, hardware firewalls or even just NAT enabled routers are as good if not better overall than the build-in software. Source: Employed in ISP/Carrier security/infrastructure 2006-2015
Should these easily reverted steps fix the problem for you, please spread the word.