Basically as much as you need. There's a lot of different figures floating around as (as best as I can tell) some take into account the height to the top edge of the side panels, some allow for fan clearance in the roof section, some measure the whole thing, and some are just conservative.
This is one of the better images to show the kind of clearance you have available, showing a good 190mm of clear space, which more or less covers you for any cooler you could want.
Your issue is that you are likely to be far more limited by your motherboard than by the case (mainly due to the ram slot location), so how much height you have is irrelevant when you potentially don't have the width. Some tower coolers may seem compatible, but will...