We were on the 3rd sailing (1st sailing of the Eastern itinerary). My wife wanted Eastern instead, so we ended up the 3rd (both 1st and 2nd were Western). However, I do have one of the specially printed menu's from the 1st sailing and I have a shirt from that sailing that is very special to me.
I think the asking for one thing special was directed to the OP. However, I will give you something that I remember from one of our sailings that was non-standard when it comes to other lines. It regards that menu I mention above. We were on the third sailing for the fantasy and we were eating our first night in Animators Palette which happened to be Lobster night. My wife is allergic to shellfish - but generally only if she eats it, so thought it would be safe. However with all the cracking the legs open and juices in the air, she had a pretty bad reaction. She left and went out to the lobby for fresh air (not dangerously gasping, but enough that she took an inhaler). Almost immediately a manager grabbed me and asked if she was okay. I told him that she needed some fresh air because of her allergy. The manager immediately went out to the lobby - he stayed at a respectable distance, but within eyesight of her. He had to leave but another CM took his place instantly. What followed for the next 30 minutes was a well choreographed tag team stream of CM's that were staying a respectable distance from her, but never for one second letting her out of sight until she came back. They wouldn't even let me follow - they insisted I stayed and watched the show and recorded it for both of us, but seeing as how I was concerned, they kept reporting back to me every few minutes on how she was. After she came back, our waiter gave her that menu as a keepsake since they had really nothing else they could give her to compensate her, but felt horrible about her experience.
The reason why I consider it so special to me though is that I have been on other cruises and just seen waiters or staff shrug off when someone leaves figuring they will be okay, or will go to sick bay or someone else will help them. Whether for insurance purposes or not, these people were genuinely concerned about my wife and there was never one second of doubt that even if it was not an insurance concern they would have done the same thing because they truly cared. Insisting I stay for the show and keeping me updated so I never felt concerned about her was just icing on the cake - these were not people doing a job - these were good people taking care of their fellow human beings. For that, i will always be grateful to DCL.