I'm still amazed by how easy it can (sometimes!) be to pick up at least the general gist of something that is written in another Indo-European language, even if you don't otherwise know said language.
It only really applies to written text for obvious reasons, but it can work amazingly well there. For instance, take this tidbit from the Italian Wikipedia's entry on a certain proverb:
Non c’è due senza tre è un noto proverbio della cultura popolare italiana. [...] L’idea alla base di questo proverbio è che se un evento non è unico, ovvero si ripete almeno due volte, molto probabilmente si ripeterà ancora.
Just read it a few times, and I'm sure you'll understand the gist of it even without knowing either the proverb or any Italian.
I sometimes wonder what new Indo-European languages feel like for those who already speak seven or eight reasonably fluently.