Inclusion Criteria:

1. Signed written informed consent must be obtained and documented according to International Conference on Harmonization (ICH)- Good Clinical Practice (GCP), the local regulatory requirements, and permission to use private health information in accordance with the Health Insurance Portability and Accountability Act (HIPPA) prior to study-specific screening procedures
2. A histologically or cytologically confirmed ovarian, breast, non-small cell lung, melanoma, gastric/GEJ/esophageal or other type of advanced cancer that is metastatic, unresectable, or recurrent and for which weekly paclitaxel is an acceptable therapeutic option.
3. Patients with epithelial ovarian, fallopian tube, or primary peritoneal cancer must also meet the following criteria: a. Must be either platinum-resistant or platinum-refractory according to the following definitions:(1)Platinum-resistant: a response to platinum therapy followed by progression within 6 months after completing therapy (2)Platinum-refractory: best response of stable disease or progression during platinum therapy; b. Must have had prior systemic treatment with a taxane; c. Must have received no more than 4 prior systemic cytotoxic regimens
4. Patients with melanoma must also meet the following criteria: a. If melanoma is BRAF wild-type or has BRAF mutations that are not amenable to BRAF inhibitor therapy, and the patient is a candidate for immunotherapy, must have received ipilimumab; b. If melanoma is positive for the V600E or V600K BRAF mutation, must have received at least one line of prior therapy with a BRAF-specific inhibitor; either alone or in combination.
5. Patients with triple negative breast cancer (estrogen receptor-negative (ER-), progesterone receptor-negative (PR-), and human epidermal growth factor receptor 2-negative (Her2-) must also meet the following criteria: a. Must have received at least one prior chemotherapy regimen for locally advanced or metastatic disease; b. Must have received prior taxane therapy.
6. Patients with NSCLC (adenocarcinoma, squamous, or adenosquamous histopathology) must also meet the following criteria: a. Must have disease that is stage IIIB, not curable by surgery or radiotherapy, or stage IV; b. Must have received at least one prior chemotherapy regimen for locally advanced or metastatic disease; c. EGFR-positive or ALK-positive patients must have received at least one line of EGFR-directed or ALK-directed therapy, respectively; d. Must have received prior taxane therapy.
7. Patients with adenocarcinoma arising from the esophagus, gastroesophageal junction, or stomach must also meet the following criteria: a. Must have received prior treatment with a platinum/fluoropyrimidine-based therapy with or without an anthracycline in the metastatic setting; or, in the adjuvant setting if recurrence occurred within 6 months of completing systemic adjuvant treatment; b. Patients with HER2 positive tumors must have had prior treatment with a Her2 inhibitor (e.g. trastuzumab or lapatinib); c. Patients who have received prior taxane therapy may be enrolled.
8. ≥ 18 years of age
9. Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST 1.1, see Section 9)
10. Karnofsky performance Status ≥ 70% (Section 15)
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 dose
12. Females of childbearing potential must have a negative serum pregnancy test
13. Aspartate transaminase (AST) and alanine transaminase (ALT) £1.5 × upper limit of normal (ULN), or ≤ 2.5 × ULN with metastatic liver disease
14. Hemoglobin (Hgb) ≥ 10 g/dl
15. Total bilirubin £ 1.5 × ULN
16. Creatinine £ 1.5 ´ ULN or creatinine clearance > 60 mL/min/1.73 m2 for patients with creatinine levels above institutional normal
17. Absolute neutrophil count ³ 1.5 x 109/L
18. Platelets ≥ 100 x 109/L
19. Life expectancy ≥ 3 months

Exclusion Criteria:

1. Anti-cancer chemotherapy, radiotherapy, immunotherapy, or investigational agents within 7 days of first dose provided all treatment-related adverse events have resolved or have been deemed irreversible, with the exception for a single dose radiation up to 8 Gray (equal to 800 RAD) with palliative intent for pain control up to 7 days before beginning the administration of BBI608.
2. Surgery within 4 weeks prior to first dose
3. Any known symptomatic brain metastases requiring steroids. Patients with treated brain metastases must be stable for 4 weeks after completion of that treatment, with image documentation required. 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
4. Pregnant or breastfeeding
5. Significant gastrointestinal disorder(s), in the opinion of the Principal Investigator, (e.g., Crohn's disease, ulcerative colitis, extensive gastric and small intestine resection)
6. Unable or unwilling to swallow BBI608 capsules daily
7. Uncontrolled intercurrent 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), uncontrolled infection or psychiatric illness/social situations that would limit compliance with study requirements
8. Known severe hypersensitivity to paclitaxel
9. Abnormal ECGs (ie, QT prolongation - QTc > 480 msec, signs of cardiac enlargement or hypertrophy, bundle branch block, signs of ischemia or necrosis and Wolff Parkinson White patterns)