In Victarions first chapter in dance, he comments that the Volantenes were doubting between sending 300 or 500 warships to Meereen. Given that Braavos is more powerful, and that it has been made clear that they are particularly efficcient at building warships, how many warships can Braavos throw in a war? 700?

Well, given that Braavos’ Arsenal can build 365 warships in a year, it depends how many years the Arsenal is put into action leading up to a war, how long the ships last (historically, wooden ships tended to last only 20-30 years), their rate of construction vs. loss*, and how much manpower they have available to crew them. 

I would guess 700 at a minimum. 

*this raises another issue: Braavos’ naval might could be not so much how many ships they have but how quickly they can replace them. The Romans, for example, didn’t have the biggest armies and often not the most armies, but they had an ability to raise new armies that no one else could compete with.