The purpose of this study is to find out what effects, good or bad, the combination of standard chemotherapy agent paclitaxel with the investigational (experimental) drug afatinib that targets HER2, has on HER2-positive esophagogastric cancer that started to get bigger despite previous treatment with trastuzumab. The doctors will also study the tumor to understand why it grew while on trastuzumab treatment and to see the effects afatinib and paclitaxel has on the tumor.