I know this post is old now, but it's the first thing that pops up on Google when trying to find a solution for a lost product key. I thought it might be useful for anyone who has this problem later on.
I too, lost my product key - lost the whole game in fact in the move. But, I had the game installed on the computer and was planning on transferring it to origin (Which you need the product key for)
Step1: Open the windows start bar and search 'regedit'
Step 2: In the directory, expand SOFTWARE -> WOW6432Note (Will probably be different on your compyer) -> Maxis -> SimCity.
Step 3: To the right, all product files should appear, including that magical code.
Step 4: Copy and paste, or write down on another piece of paper
Step 5: Congratulations, you now have the product key again.
Hope this helps any lost soul.