Once you are onboard, read through your daily compass every evening, they sometimes run specials for laundry...That is the only major fault I find with Royal Caribbean, no self service laundry rooms, sure wish they would add some of those....
Sorry, the link seems to be working very weird, you have to save it to your computer and then open it (maybe as it is a PDF?), you can't just open it directly (at least I can't).
As Donna said, there are sometimes "wash and fold" days with specials for laundry.
I can understand that it would be convenient to have self service laundry if you for example have kids, but I prefer the laundry service. Even if there are a couple of sea days during a cruise I usually don't have time to do the laundry myself (too much to do onboard and why waste time in a laundry room... ).
On most cruises (usually 5 night or more) Royal Caribbean will have a 'wash and fold special' at least once where everything you can fit in a plastic bag they leave on the bed will be done for a flat fee. I want to say it is $20. It is not a very large bag but it does stretch a bit. I've been able to get a pretty surprising amount of stuff in it. Roll stuff tight!