Pleco is one essential for me, although it's obviously only useful in Chinese-speaking countries.
It's free (with extra dictionaries and bits that can be paid for), and allows translation to and from Chinese (Traditional and Simplified).
You can enter Chinese characters by building them up from components, or drawing them on the screen (although they have to be drawn in the right stroke sequence, which means you need some knowledge of the script).
It's not really perfect for quick translations when you need to buy a cup of coffee - but it's excellent for helping while you are learning the language or curious about the meaning of a sign.
The only other essential travel app I'd mention is a currency convertor - but there's plenty around, so I don't know particularly which one to recommend.
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