EDIT 2: It seams that launching the multiplayer trough google Chrome adds/is responsible/, for the problem ( again when i try to enter a multiplayer game i got the mesage inside Origin client stating that Back to Karkand is not installed).
Use mozilla or diferent browser than chrome.
EDIT 1: It appears that disabling, "Enable Origin In Game for Battlefield 3™ Limited Edition", inside origin client leads to this.
If the game is installed i got eror mesage stating that Back to Karkand expansion is not installed ( i got Limited Edition).
For now alowing the In game Origin function does the trick.
The problem is cause be stalled/stuck webhelp process - once u try to delete the "C:\ProgramData\Origin\", there is a process hodling the "C:\ProgramData\Origin\ServiceLogs" folder that u cant delete (ive used a 3rd party program to unlock the folder and saw the process hodling the folder webhelper.exe).
Also for me worked the web start method - u open origin.com login, click on your game (BF3 in my case), open with origin and from there i was able to start the download process. It happend today on a fresh win 10 pro installed and updated. At the begining of initial origin install, ive encouterd a msvcp 140 (something) error, related to missing vc 2015 redistribuatbles.