Great question and good point about focusing on the app itself. There are more and more BAAS companies every day, and it might be a good idea to consider them, although I haven't looked into their prices.
On the other hand, if you want to prototype fast, and cheap in the same time, why don't you just forget about the scalability issues of your backend altogether? After all, prototyping is all about iterating, with minimal server load. Any cheap (or even free) LAMP server will do. Write minimal throwaway backend code, and finalize when getting close to market fit.
The quality of your backend will be very important later on, but while prototyping, I don't see a reason why wouldn't you get away with a simple default php-mysql setup. For many usecases, even a textfile will do as database in the early days. :)
That said, I'm not a professional developer, so I might have overlooked some aspects here.