A Study of BBI503 in Combination With Selected Anti-Cancer Therapeutics in Adult Patients With Advanced Cancer

Major inclusion criteria: 1. A histologically or cytologically confirmed solid tumor that is locally advanced, recurrent, or metastatic; for which curative resection is not currently possible; and for which systemic treatment with one of the selected anti-cancer agents is a reasonable therapeutic option. 2. Must be ≥ 18 years of age 3. Has disease such that progression or response to therapy can be evaluated objectively while on protocol. 4. Must have Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 5. Male or female patients of childbearing potential must agree to use contraception or avoidance of pregnancy measures during the study and for 30 days after the last dose. 6. Females of childbearing potential must have a negative serum pregnancy test. 7. Must have aspartate transaminase (AST) ≤ 2.5 × upper limit of normal (ULN) and alanine transaminase (ALT) ≤ 2.5 × ULN. Patients who do not have hepatocellular carcinoma but who have liver lesions or liver metastases may be eligible if AST ≤ 3.5 × ULN and AST ≤ 3.5 × ULN if agreed upon by the investigator and medical monitor for the sponsor. 8. Hemoglobin (Hgb) ≥ 9 g/dl 9. Total bilirubin ≤ 1.5 × ULN. For patients with liver lesions, total bilirubin ≤ 2.0 × ULN may be enrolled if agreed upon by the investigator and medical monitor for the sponsor 10. Creatinine ≤ 1.5 × ULN or creatinine clearance ≥ 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional upper limit of normal (using the Cockcroft-Gault equation). 11. Absolute neutrophil count ≥ 1.5 × 10\^9/L 12. Platelets ≥ 100 × 10\^9/L 13. Life expectancy ≥ 3 months Major exclusion criteria: 1. Received anti-cancer chemotherapy, radiotherapy, immunotherapy, or investigational agents within 7 days of first dose of protocol therapy. Patients may begin protocol therapy on a date determined by the investigator and medical monitor for the sponsor after a minimum of 7 days since last receiving anti-cancer treatment, provided that all adverse events related to that have resolved or have been deemed irreversible. 2. Major surgery within 4 weeks prior to first dose; major surgery is defined as a procedure requiring any of the following: general anesthesia, intubation and mechanical ventilation, or major incision (e.g., thoracotomy, laparotomy) 3. Any known, untreated, brain metastases. Patients with treated brain metastases must have no clinical symptoms from the metastases, and must be either off steroids or on a stable dose of steroids ≤ 10 mg prednisone or equivalent for at least 2 weeks prior to protocol enrollment. Patients with known leptomeningeal metastases are excluded, even if treated. 4. Pregnant or breastfeeding or expecting to conceive or father children within the projected duration of the study. 5. Significant gastrointestinal disorder(s), in the opinion of the Principal Investigator, such as active inflammatory bowel disease, extensive gastric or small intestinal resection (which has resulted in short-gut syndrome or the inability to take oral medications). 6. Unable or unwilling to swallow either BBI503 daily or an oral selected anti-cancer therapeutics; or, unwilling to receive intravenous injection of IV anti-cancer therapeutics. 7. Positive for Human Immunodeficiency Virus (HIV), Hepatitis B (Hepatitis B Surface Antigen \[HBsAg\] reactive), or Hepatitis C virus (Hepatitis C Virus Ribonucleic Acid (HCV RNA\] (qualitative) is detected). 8. Uncontrolled concurrent illness including, but not limited to: ongoing or active infection requiring therapy, 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 on mild exertion), uncontrolled infection or psychiatric illness/social situations that would limit compliance with study requirements 9. 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; c) localized prostate cancer not requiring systematic therapy; and d) other primary cancer with no known active disease present, and no treatment administered in the 2 years prior to enrollment. 10. For patients to be treated with a regimen containing capecitabine: a) Known hypersensitivity to capecitabine, b) Known dihydropyrimidine dehydrogenase (DPD) deficiency, c) Significant gastrointestinal disorder(s) that would, in the opinion of the Investigator, prevent absorption of an orally available agent 11. For patients to be treated with a regimen containing sunitinib: a) Uncontrolled hypertension (systolic blood pressure \> 150 mmHg or diastolic pressure \> 90 mmHg despite optimal medical management), b) Evidence of bleeding diathesis or a clinically significant coagulopathy (≥ CTCAE Grade 3) within 4 weeks prior to the start of study, c) Recent hypoglycemia, d) Uncontrolled thyroid dysfunction despite optimal medical therapy 12. For patients to be treated with a regimen containing doxorubicin: a) Known left ventricular ejection fraction \< 50%, b) Hypersensitivity to doxorubicin 13. A patient to be treated with a regimen containing nivolumab or pembrolizumab will be excluded if the patient: a) Has an active autoimmune disease requiring immunosuppression with the exception of subjects with isolated vitiligo, resolved childhood asthma or atopic dermatitis, controlled hypoadrenalism or hypopituitarism, and euthyroid patients with a history of Grave's disease, b) Has had a previous life-threatening (CTCAE grade 4) immune-mediated adverse reaction; or, a previous severe (CTCAE grade 3) immune mediated adverse reaction that required treatment with corticosteroids (more than 10 mg/day prednisone or equivalent dose) for longer than 12 weeks, c) Has a transplanted organ, d) Has interstitial lung disease or active, non-infectious pneumonitis, e) Has received a live vaccine within 30 days prior to first dose, f) Previous severe hypersensitivity reaction to another monoclonal antibody (mAb), g) Has been treated with another monoclonal antibody ≤ 4 weeks before first dose.