Just to add my own comments to this thread since I experienced the 0x800f0906 error myself a couple of days ago when DISM failed to resolve the issue.
In my particular case, the machine was running perfectly until I installed a number of Optional updates which had appeared the day before. These were as follows:
https://support.microsoft.com/en-us/kb/3029432
https://support.microsoft.com/en-us/kb/3029603/
https://support.microsoft.com/en-us/kb/3034348
https://support.microsoft.com/en-us/kb/3041857
https://support.microsoft.com/en-us/kb/3045634
https://support.microsoft.com/en-us/kb/3045746
https://support.microsoft.com/en-us/kb/3054464
https://support.microsoft.com/en-us/kb/3055999/
I didn't install KB3029603 since I have an Intel processor and neither did I install KB3045634, KB3055999 and KB3029432. But I did install the others even though I wasn't experiencing the issues raised in the appropriate articles.
A reboot was required which went according to plan, but later the same day, the machine wouldn't shutdown anymore and had to be manually powered off. It was after that that I decided to run SFC which couldn't fix the errors followed by DISM which has always
worked in the past, but failed this time around.
I tried running the suggestion on the WindowsClub site which is this one: http://www.thewindowsclub.com/repair-corrupt-windows-image-error-0x800f0906 but that failed to resolve the issue as well.
Since the only installations that had been performed that day were the optional updates, I decided to run System Restore to a time prior to installing them. That fixed the problem I'm glad to say.
The lesson here I think is not to install everything presented as an optional update even though one of them may well prevent a problem occurring at some time in the future.
Also, running System Restore has gotten me out of trouble on more than one occasion and it's well worth a try regardless of the issue.