A Study of BBI608 in Combination With Sorafenib, or BBI503 in Combination With Sorafenib in Adult Patients With Hepatocellular Carcinoma

Inclusion Criteria 1. Signed written informed consent must be obtained and documented according to International Conference on Harmonisation (ICH) and local regulatory requirements 2. Histologically or cytologically confirmed hepatocellular carcinoma that is metastatic, unresectable, or recurrent. 1. Patients must not be candidates for curative resection 2. Patients who have recurrent disease after having had one or more prior resections may be eligible, provided that they are not candidates for further curative resection. 3. Patients who have recurrent hepatocellular carcinoma following hepatic transplantation are excluded unless the following criteria are met: i. Transplantation was performed at least 6 months prior to the relapse of HCC. ii. Patients are on stable immune suppressive therapy with no clinical evidence of rejection. iii. Are receiving ≤ 2.5 mg everolimus daily. d. Patients with known HIV infection are excluded. e. Patients with Hepatitis B are eligible provided there is no active viral replication. Patients with Hepatitis C who are not on interferon are eligible. 3. Patients who have a diagnosis of hepatocellular carcinoma made through radiologic imaging may be eligible, provided they meet the criteria according to the American Association for the Study of Liver Disease, AASLD (Bruix and Sherman, 2005; Bruix and Sherman, 2011) 4. Patients must be candidates for sorafenib 5. Must have had no previous systemic anti-cancer treatment, though previous loco-regional therapy is allowed: a. Prior treatment with any of the following is allowed: trans-arterial embolization, trans-arterial chemo-embolization, percutaneous ethanol injection, radio-embolization, radio-frequency ablation, or other ablation techniques. 6. Must be Child-Pugh class A a. Patients with uncontrolled massive ascites or presence of hepatic encephalopathy are excluded 7. Must have total serum bilirubin ≤ 3 mg/dl 8. ≥ 18 years of age 9. Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 10. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 11. Male or female patients of child-producing potential must agree to use contraception or avoidance of pregnancy measures during the study and for 30 days after the last BBI608 or BBI503 dose 12. Females of childbearing potential must have a negative serum pregnancy test 13. Aspartate Aminotransferase (AST) and Alanine transaminase (ALT) \< 5.0x the upper limit of normal (ULN) 14. Glomerular filtration rate (GFR) \> 45 mL/min/1.73m\^2 according to the Cockcroft-Gault estimation. 13\. Hemoglobin ≥ 8.5 mg/dl 14. Absolute neutrophil count ≥ 1.5 x 10\^9/L 15. Platelets ≥ 75 x 10\^9/L 16. Life expectancy ≥ 3 months Exclusion Criteria 1. Previous treatment with sorafenib 2. Patients with known hypersensitivity to sorafenib or any other component of sorafenib. 3. Previous systemic anti-vascular endothelial growth factor (VEGF) or any prior systemic anti-cancer therapy, including prior treatment with systemic agents such as regorafenib, ramucirumab, pazopanib, or experimental agents such as brivanib. 4. Have had a surgical procedure requiring general anesthesia or inpatient hospitalization for recovery less than 4 weeks prior to beginning protocol therapy. 5. Have had a loco-regional procedure for the treatment of hepatocellular carcinoma (such as a percutaneous, trans-arterial, or radio-ablative procedure) less than 4 weeks prior to beginning protocol therapy. Protocol therapy may begin a minimum of 4 weeks after such a procedure provided the following criteria are met: 1. There is progression of disease documented by RECIST 1.1 2. All adverse events from the procedure have resolved or have been deemed irreversible and the patient meets inclusion criteria. 6. Any known symptomatic or untreated brain metastases requiring increase of steroid dose within 2 weeks prior to starting on study. Patients with treated brain metastases must be stable for 4 weeks after completion of that treatment. Patients must have no clinical symptoms from brain metastases and must be either off steroids or on a stable dose of steroids for at least 2 weeks prior to protocol enrollment. Patients with known leptomeningeal metastases are excluded, even if treated. 7. Pregnant or breastfeeding 8. Significant gastrointestinal disorder(s), (e.g., Crohn's disease, ulcerative colitis, extensive gastric and small intestine resection) such that, in the opinion of the treating investigator, absorption of oral medications may be impaired. 9. Unable or unwilling to swallow BBI608, BBI503, or sorafenib capsules or tablets 10. Uncontrolled inter-current illness including, but not limited to: ongoing or active infection, clinically significant non-healing or healing wounds, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, significant pulmonary disease (shortness of breath at rest or mild exertion), or uncontrolled infection or psychiatric illness/social situations that would limit compliance with study requirements (e.g. no reliable transportation). 11. Subjects with a history of another primary cancer, with the exception of: a) curatively resected non-melanoma skin cancer; b) curatively treated cervical carcinoma in situ; or c) other primary solid tumor with no known active disease present that, in the opinion of the investigator, will not affect patient outcome in the setting of current hepatocellular carcinoma diagnosis. 12. Abnormal ECGs which are clinically significant such as QT prolongation (QTc \> 480 msec), clinically significant cardiac enlargement or hypertrophy, new bundle branch block, or signs of active ischemia. Patients with evidence of prior infarction who are New York Heart Association (NYHA) functional classes II, III, or IV are excluded, as are patients with marked arrhythmias such as Wolff Parkinson White pattern or complete atrioventricular (AV) dissociation.