I'm also having this issue. Fancied one more quick game before the last 30 minutes of my Game Time expired, only for Origin to bug me about a product key. While it's only 30 minutes for me, it's still not an acceptable circumstance for potential customers to be screwed out of any time, accidentally or otherwise. What's doubly annoying, though, is the client then pops the "You still have free time left!" nag screen after you close the activation window.
Have some pretty pictures to prove I'm not just pulling this out of the air! :D
http://i946.photobucket.com/albums/ad303/Idsertian/OriginGameTime1_zps377c840c.png
http://i946.photobucket.com/albums/ad303/Idsertian/OriginGameTime2_zps23c1b8f8.png
Question: What are you going to do about this, EA? Irritated users wish to know...
