The word for "fun" exists in both German and English, but is missing from many other languages.
Many social constructs don't translate well. The word "freunde" is much, much stronger than "friends", for example. "We're all friends here" doesn't really work in German, at least not in the same situations. Similarly, English doesn't really have a term for "Wohnunggemeinschaft."
Puns often don't translate, which breaks translating many of the jokes in Shakespeare, for example. Going from German to English, the Rammstein song "Du has(s)t much" doesn't translate due to the puns, as does much of the writing by Yoko Tawada.
Gender ambiguous things don't always translate either. The gender neutral pronoun "they" translates to "sie" in German, which is the same as the pronoun "she" but with different verb conjugations. "Trans man" or "transwoman" are usually translated in German the same as "man" or "woman". Nuance on racial issues also doesn't always translate smoothly between the two languages.
I am sure there are many, many more.