That's from condensation, isn't it? My place has the same problem - and it comes back (and/or springs up in new areas) after cleaning with bleach spray, because the condensation problem remains. Is there an extractor fan in the bathroom? If not, might be worth fitting one, whatever other decisions you make. For what it's worth, when I was renting I'd have fully expected to be chucked out if I didn't clean it after being asked twice. At some point it'll be bad enough that you'll need to re-paint rather than just clean. And at some point after that, it'll be bad enough that you'll need to re-plaster. You have to draw a line somewhere, right?