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. Patients with thymic carcinoma must have received at least one prior systemic
             chemotherapy regiment for metastatic, recurrent, locally advanced or otherwise
             unresectable disease.

          9. ≥ 18 years of age

         10. Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST
             1.1, see Section 9)

         11. Karnofsky performance Status ≥ 70% (Section 15)

         12. 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

         13. Females of childbearing potential must have a negative serum pregnancy test

         14. Aspartate transaminase (AST) and alanine transaminase (ALT) £1.5 × upper limit of
             normal (ULN), or ≤ 2.5 × ULN with metastatic liver disease

         15. Hemoglobin (Hgb) ≥ 10 g/dl

         16. Total bilirubin £ 1.5 × ULN

         17. Creatinine £ 1.5 ´ ULN or creatinine clearance > 60 mL/min/1.73 m2 for patients with
             creatinine levels above institutional normal

         18. Absolute neutrophil count ³ 1.5 x 109/L

         19. Platelets ≥ 100 x 109/L

         20. 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)