This in a nutshell is the biggest difference between Disney Cruise Line and all others, because only Disney can "do Disney." For cruises with a lot of sea days, I enjoyed them the most on my Disney Magic Transatlantic cruise last May. I am looking forward to 15 nights on the re-imagined Magic's Transatlantic in May 2015.
Last year we went on a Mediterranean cruise on DCL. We stopped in Rome on the last Sunday of the month. The Vatican museum is free to the public but closed in the afternoon. DCL was able to bring docents and security guards in the afternoon and had the entire museum to ourselves. When the Wonder and Magic first came out, I remember a trivia question stating that DCL lifeboats were painted yellow to match Mickeys shoes, where all other boats were a Coast Guard orange. That is some serious influence.