Let $ n $ be the number of matches she won before the weekend began. Since her win ratio started at exactly .$ 500 = \tfrac{1}{2}, $ she must have played exactly $ 2n $ games total before the weekend began. After the weekend, she would have won $ n+3 $ games out of $ 2n+4 $ total. Therefore, her win ratio would be $ (n+3)/(2n+4) $. This means that \[\frac{n+3}{2n+4} > .503 = \frac{503}{1000}.\]Cross-multiplying, we get $ 1000(n+3) > 503(2n+4), $ which is equivalent to $ n < \frac{988}{6} = 164.\overline{6} $. Since $ n $ must be an integer, the largest possible value for $ n $ is $ \boxed{164} $.