Hey guys.
I had the exact same problem Sentinel described down to a T and spent hours with EA online tech support staff (3 of them) before getting the problem solved by the last guy. Thumbs up for Joseph 🙂 I was missing my Product Code (For reasons unknown - Likely a Origin glitch).
If you open up your Origin and go to your games library you should be able to get your "Product Code" by right clicking on your games and selecting "Show Game Details". If the game doesn't show a Product Code then that is where the issue lies. If you don't have the Product Code then everything seems to go straight to hell when trying to get the game to work (Including steps in other links for the same problem).
After establishing that I did not have a product code for ME (Though I did for all my other games including ME 2 and ME 3 which I also purchased at the same time), the EA game tech got me to exit origin whilst he reset my game details. All I had to do was follow a link and inuput a game code that the Tech gave me and then download the game again (It could be worse right? Hah).
If you have the same issue that I did, ensure that you contact EA support chat through the below link or an out of origin link.
https://help.ea.com/uk/contact-us/contact/chat
If you contact them through the Origin client, when you close the client as one of the steps to fix the issue you will also lose your live chat.
Really hopes this helps others because It was frustrating as hell for me 🙂