A button-up that needs an iron is a bad travel shirt. Bluffworks calls the Meridian machine washable and wrinkle free, and says it needs no ironing and no dry cleaning. The fabric is stretch polyester.
This Sunrise Plaid classic fit lists $128. The page also describes a hidden pocket and a zippered pocket behind it.
Shortlist reason: a button-down the maker calls wrinkle free and machine washable.