Easy answer, Black Friday Walmart, Laptop for $173. Otherwise anything else is more expensive then $200 because buying Windows alone is $119 (http://www.microsoftstore.com/store/msusa/en_US/pdp/Win...). The next best offer would be to get a i3 Core Laptop or Desktop (seen some desktop come with free LCD in the box too) for $249 at Walmart. Again gaming would be ONLY LOW, you would have to spend significantly more to get medium settings (and again depends what game you talking about Medium in BF4 or you talking FarmVille?).
i use a stock i3 laptop for gaming when im not at my desktop and i play world of warcraft at mid settings with 25-35 fps mostly so it would do fine. if you can get one cheap on black friday otherwise my build and a copy of windows from a friend is the best bet