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.

Inclusion Criteria:

• Pathologically or cytologically confirmed esophagogastric cancer.
• HER2 overexpression and/or amplification as determined by immunohistochemistry (3+) or FISH (≥2.0)
• Previously received trastuzumab as part of a regimen in the perioperative or metastatic setting with evidence of progression 9Zr-trastuzumab use as imaging agent for 89Zr-trastuzumab PET permitted..
• May have previously received lapatinib as part of a regimen in the perioperative or metastatic setting with evidence of progression of disease. Washout period for lapatinib of 14 days.
• Completion of previous chemotherapy regimen ≥2 weeks prior to the start of study treatment. Other chemotherapy regimens may have been administered between the time of progression on prior trastuzumab containing regimen and protocol therapy. No restriction on prior chemotherapy regimens for advanced stage disease.
• At least one measurable metastatic lesion according to RECIST 1.1 criteria. Ascites, pleural effusions, and bone metastases are not considered measurable. Minimum indicator lesion size = 10 mm by helical CT or = 20 mm by conventional techniques. Pathological nodes must be = 15 mm by the short axis to be considered measurable.
• Patients aged 18 years or older, as no dosing or adverse event data are currently available on the use of afatinib in patients <18 years of age, children are excluded from this study.
• Life expectancy of at least three (3) months.
• Karnofsky performance status ≥60%
• All patients with disease technically amenable to biopsy will be asked to undergo a biopsy. Patient must agree to allow 2 biopsies of any malignant lesion that can be accessed by endoscopy or with the aid or radiology (i.e. CT guided).
• Patients who have previously provided samples at any time after trastuzumab resistance will be exempt from biopsy at the start of therapy.
• Consent to preservation of frozen and fixed samples of tumor cores for evaluation
• Able to swallow and retain oral medication.
• Negative serum HCG pregnancy test for premenopausal women of reproductive capacity and for women less than 12 months after menopause.
• Willingness to use birth control while on study.
• Asymptomatic, central nervous system metastases are permitted.

Exclusion Criteria:

• Patients receiving any concurrent anticancer therapy or investigational agents with the intention of treating esophagogastric cancer. 89Zr-trastuzumab uses as imaging agent for 89Zr-trastuzumab PET permitted.
• Prior disease progression on docetaxel or paclitaxel in metastatic setting.
• Patients who are unwilling to consent to mandatory tumor biopsy. Patients with archival tissue permitted to enroll on study per MSK Principal Investigator discretion Women who are pregnant or breast feeding.
• Concurrent radiotherapy is not permitted for disease progression on treatment on protocol (except in the context specified in section 9.0), but might be allowed for pre-existing non-target lesions with approval from the principal investigator of the trial.
• Concurrent medical conditions which may increase the risk of toxicity, including ongoing or active infection, history of significant bleeding disorder unrelated to cancer (congenital bleeding disorders, acquired bleeding disorders within one year), HIV-positive.
• Subjects with acute Hepatitis B are not eligible. Subjects with chronic hepatitis are eligible if their condition is stable and in the opinion of the investigator, if consulted, would not pose a risk to subject safety.
• History or presence of clinically relevant cardiovascular abnormalities such as uncontrolled hypertension, congestive heart failure NYHA classification of 3, unstable angina or poorly controlled arrhythmia. Myocardial infarction within 6 months prior to study entry.
• Baseline (< 1 month before treatment) cardiac left ventricular function with resting ejection fraction of less than 50% measured by echocardiogram.
• Known pre-existing interstitial lung disease.
• Significant or recent acute gastrointestinal disorders with diarrhea as a major symptom e.g., Crohn's disease, malabsorption, or CTCAEGrade >2 diarrhea of any etiology.
• Unwillingness to give written informed consent, unwillingness to participate, or inability to comply with the protocol for the duration of the study.
• Active hepatitis B infection, active hepatitis C infection
• Known HIV carrier
• Known or suspected active drug or alcohol abuse. Restricted Therapies
• Additional experimental anti-cancer treatment and/or standard chemo-, immunotherapy, hormone treatment (with the exception of megestrol acetate), or concurrent radiotherapy is not allowed concomitantly with the administration of study treatment (with the exception listed in section 9.0) 89Zr-trastuzumab use as imaging agent for 89Zr-trastuzumab PET permitted. Afatinib is a substrate of P-gp and its plasma concentrations can be affected by the use of P-gp inhibitors (data on file) and it is also likely that P-gp inducers could also influence afatinib plasma concentrations.
• The use of potent P-gp inhibitors (including cyclosporine, erythromycin, ketoconazole, itraconazole, quinidine, Phenobarbital salt with quinidine, ritonavir, valspodar, verapamil) and potent P-gp inducers (including St John's wort, rifampicin) has to be avoided during treatment with afatinib. Any exemptions to this have to be discussed with the principal investigator.