On my last cruise - aboard the Oasis Of The Seas - my bartender friend Reynald made me his "special" rumrunner. He squeezed juices from limes and oranges (no mixes or pre-made mixers), carefully added various rums and other things, and it was really the best cocktail I've ever drank.
I guess if I had to pick just one it would be a BBC made with real bananas. Like Dave, I've made friends with a few ship bartenders over the years and they'll make drinks for me from fresh ingredients sometimes. I don't usually drink frozen drinks very often..but man, that hit the spot after walking all day in the heat.
For me.... its a drink called kiss on the lips. I get a kick out of ordering it from the waiters. Its made with peach schnapps, grenadine syrup, and mango mix.
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Fascination-September 2007 Key West and Cozumel
Spirit-August 2010- Alaska
It may not be classified as a "drink", but I always enjoy a hot cup of coffee with hot chocolate mixed in every morning on a cruise.
MamaofEnS
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RCI Radiance of the Seas Sep 2009
NCL Sun March 2008
Carnival Destiny June 2006
Carnival Ecstasy June 2002
Carnival Elation August 2000
Celebrity Century June 1998
Carnival Carnivale August 1993
MamaofEnS, I'm kind of with you. My favorite drink on a cruise is a cup of tea, taken on an outside deck just before sunrise.
As for adult beverages, I almost always enjoy a Cosmopolitan as my first drink of the cruise. The best one was on SS Norway, in Club Internationale. Must have been the setting.