Brief Summary: This research study is evaluating Lorlatinib in combination with Crizotinib, Binimetinib, or TNO155 as a possible treatment for either anaplastic lymphoma kinase (ALK)-positive lung cancer or ROS1-positive lung cancer. * This research study involves four study drugs. * Lorlatinib * Binimetinib * Crizotinib * TNO155

Detailed Description: This is a Phase I/II clinical trial of three investigational combinations for treatment of either anaplastic lymphoma kinase (ALK)-positive or ROS1-positive lung cancer. The three drug combinations being tested are (1) Lorlatinib combined with Crizotinib and (2) Lorlatinib combined with Binimetinib and (3) Lorlatinib combined with TNO155. * Lorlatinib is an oral ALK and ROS1 inhibitor. The US Food and Drug Administration (FDA) has approved Lorlatinib for treatment of ALK-positive lung cancer. The FDA has not approved Lorlatinib for treatment of ROS1-positive lung cancer. * Crizotinib is an oral ALK and MET inhibitor. The FDA has approved Crizotinib for treatment of ALK-positive lung cancer. Crizotinib is not approved by the FDA for the treatment of MET-positive lung cancer. This study will test crizotinib's ability to block MET signaling. Crizotinib is not approved by the FDA for treatment of ROS1-positive lung cancer. * Binimetinib is an oral MEK inhibitor. The FDA has not approved binimetinib for treatment of ALK-positive or ROS1-positive lung cancer but it has been approved for other uses. * TNO155 is an oral SHP2 inhibitor. The FDA has not approved TNO155 for the treatment of any disease, including ALK-positive or ROS1-positive lung cancer. It is an investigational drug. * The FDA has not approved the combination of Lorlatinib with Binimetinib, Crizotinib, or TNO155 as a treatment for any disease. The research study procedures include screening for eligibility and study treatment which will include evaluations and follow up visits. * Patients will undergo screening and those who fulfill the eligibility criteria will be assigned to receive either the combination of Lorlatinib and Crizotinib, the combination of Lorlatinib and Binimetinib, or the combination of Lorlatinib and TNO155. Patients with ALK-positive or ROS1-positive lung cancer who have extra copies of the MET growth signal (MET amplification) will be assigned to receive the Lorlatinib and Crizotinib combination. All other patients will be randomly assigned to receive one of the combinations. * This study consists of 2 parts: * Phase I: * The investigators are looking to determine whether combining Lorlatinib with either Binimetinib, Crizotinib, or TNO155 is well tolerated and to understand how treatment with the two drugs affects cancer cells and impacts growth signals. * Not everyone who participates in this research study will receive the same dose of the study drug. The dose given will depend on the number of participants who have been enrolled in the study prior and how well the dose was tolerated. * Phase II: * The Phase II study will test the safest doses of the combinations (as established in the Phase I study) in a larger group of patients. * It is expected that about 96 people will take part in this research study