It's true that each EOD can be refinished once without charge. The watch must be returned to Switzerland and disassembled for this. I had one serviced at the same time I had this done and do not know whether service was required. It did take a surprisingly long time for the watch to return, which is unusual for AP service, but it looked like new, which is typical of AP.
Park

I was not the first owner of the one I had re-done. I simply took it to the AP Boutique (in my case, in New York) with the papers and asked them to check if it had been done previously.
Here's how it looked when it came back (photo credit to MichaelC, who took the best photo of it to date):

Sorry, forgot to add picture to last post.
Park
Hi,
The original intention of the complimentary ONE TIME refinish offer was for the original owner, purchasing through an authorized dealer, but as a courtesy, AP ended up honoring the offer even for second and later owners, if the piece had never been submitted for a refinish.
There was never an official exact expiry for the offer, at least in most markets.
If you are a first owner, purchased from an authorized dealer or boutique, and you know your piece has never been submitted for the complimentary refinish, I suggest you contact your local boutique (AP Boutique) and ask them about it. You might be surprised...
If you are NOT the first owner, purchased through an authorized boutique or dealer, you could still ask but don't be surprised to find out the offer has "expired."
One of the (possible) advantages of buying through an AD or boutique...
Cheers,
TM