I like Julien's answer but I think it a bit low
Generally I find that most native speakers get away with 6000 words, but of course they also use phrases and sayings, which require you to understand not just the words but also what the phrase means.
That will get you through your day, buy food, catch a bus, talk about the weather
The moment you want to do anything more complicated, rent a house long term, enter into a contract etc you need to up that figure above 15,000. I've just been working with a very bright Swiss who wanted to discuss a minor financial matter and he had to work hard to understand the meaning of certain words, because the words that describe them also needed looking up.