Patients must not have known hypersensitivity to eflornithine or sulindac or the excipients byproducts; patients must not have experienced asthma, urticaria, or allergic-type reactions after taking aspirin or other non-steroidal antiinflammatory drugs (NSAIDs) Patients must not be receiving or plan to receive concomitant oral or intravenous corticosteroids on a regular basis, nonsteroidal anti-inflammatory drugs (NSAIDs), nor anticoagulants on a regular or predictable intermittent basis; (NSAID use may not exceed 10 days per month); patients may receive daily aspirin for cardiovascular prophylaxis as long as acetylsalicylic acid (ASA) is =< 100 mg per day or =< two 325 mg tablets per week; inhaled steroids (i.e. for asthma or related conditions) are allowed The following concomitant medications are not allowed during navitoclax administration: clopidogrel, ibuprofen, tirofiban, warfarin, and other anticoagulants, drugs, or herbal supplements that affect platelet function are excluded, with the exception of low-dose anticoagulation medications (such as heparin) that are used to maintain the patency of a central intravenous catheter; aspirin will not be allowed within 7 days prior to the first dose of navitoclax or during navitoclax administration; however, subjects who have previously received aspirin therapy for thrombosis prevention may resume a low dose (i.e., maximum 100 mg QD) of aspirin if platelet counts are stable (>= 50,000/mm^3) through 6 weeks of navitoclax administration; all decisions regarding treatment with aspirin therapy will be determined by the investigator in conjunction with the medical monitor Ongoing use of anticoagulant therapy other than aspirin or nonsteroidal anti-inflammatory drugs (NSAIDs) that places the participant at increased bleeding risk in the opinion of the site investigator Current use of certain concomitant medications due to mechanistic-based platelet toxicities from navitoclax: clopidogrel, ibuprofen, tirofiban and other anticoagulants, drugs or herbal supplements that effect platelet function; NOTE: antiplatelet use is prohibited during the use of navitoclax; subjects who have previously received aspirin therapy for thrombosis prevention may resume a low dose (i.e., maximum 100 mg QD) of aspirin if platelet counts are stable (>= 50,000/mm^3) through 6 weeks of navitoclax administration; all decisions regarding treatment with aspirin therapy will be determined by the principal investigator in conjunction with the medical monitor Ongoing use of anticoagulant therapy other than aspirin or non-steroidal anti-inflammatory drugs (NSAIDS) that cannot be stopped for surgical procedures Anticoagulation and anti-platelet therapies are not permitted (this includes Coumadin, low molecular weight heparins, factor Xa inhibitors, aspirin and non-steroidal anti-inflammatory drug [NSAIDS] or other medicines with similar effects) Patients who are chronically receiving aspirin, ibuprofen or other non-steroidal anti-inflammatory drugs are not eligible Current cardiac arrhythmia requiring concurrent use of anti-arrhythmic drugs Patients receiving anticoagulation or anti-platelet therapy are excluded; excluded agents include heparin or low molecular weight heparin, warfarin, clopidogrel, ibuprofen and other nonsteroidal anti-inflammatory drug (NSAIDS), tirofiban, and other anticoagulants, drugs, or herbal supplements that affect platelet function; administration of heparin to keep subject's infusion lines patent is allowed; low-dose anticoagulation medications that are used to maintain the patency of a central intravenous catheter are allowed; aspirin will not be allowed within 7 days prior to the first dose of navitoclax or during navitoclax administration; however, subjects who have previously received aspirin therapy for thrombosis prevention, may resume a low dose (i.e., maximum 100 mg QD) of aspirin if platelet counts are stable (>= 50,000/mm^3) through 6 weeks of navitoclax administration; all decisions regarding treatment with aspirin therapy will be determined by the investigator in conjunction with the medical monitor Regular nonsteroidal anti-inflammatory drug (NSAID)/aspirin use at any dose (including baby aspirin) (defined as >= 5 days per week) is allowed if aspirin and/or NSAIDs are stopped for 30 days prior to study entry and throughout the study period; participants will be encouraged to use acetaminophen for minor pain and fever No known allergy to aspirin Concurrent enrollment on a non-chemotherapy treatment trial will be allowed, as long as that trial allows concurrent daily aspirin use Concomitant systemic treatment with chronic use of anti-histamine or non-steroidal anti-inflammatory drugs and other platelet inhibitory agents and patients on oral anticoagulant (e.g. warfarin)