A Study of Prexasertib (LY2606368) With Chemotherapy or Targeted Agents in Participants With Advanced Cancer

Inclusion Criteria: * Must be appropriate candidate for experimental therapy, as determined by investigator, after available standard therapies have failed * Have adequate organ function * Prior Therapies: Systemic treatments: must have discontinued previous systemic treatments for cancer and recovered from the acute effects of therapy. Participants must have discontinued mitomycin-C or nitrosourea therapy at least 42 days and have discontinued any cytotoxic therapies at least 28 days prior to study enrollment. Radiation therapy and surgery: must be completed at least 4 weeks before study enrollment * All parts except Part B, Part E2, and Part E3 dose expansion: Must have diagnosis of cancer that is advanced or metastatic * Part B dose expansion: Must have confirmed Kirsten rat sarcoma viral oncogene homolog (KRAS) wild-type colorectal cancer that is metastatic or recurrent and has failed oxaliplatin- and irinotecan-based chemotherapy or who are intolerant of irinotecan or oxaliplatin * Part E2 dose expansion: must have cancer that is advanced or metastatic and have prior documentation of a mutation of PIK3CA * Part E3 dose expansion: must have advanced or metastatic ER-negative, PR-negative, and HER-2 non-overexpressing breast cancer * Must be available during the duration of the study and willing to follow the study procedures * Parts A and B: If participant is of reproductive potential, must agree to use medically approved contraceptive precautions during the study and for six months following the last dose of study drug * Parts C, D and E: If participant is of reproductive potential, must agree to use medically approved contraceptive precautions during the study and for three months following the last dose of study drug * If the participant is a female of childbearing potential, must have had a negative serum or urine pregnancy test within 14 days of the first dose of study drug and must not be breast feeding * Part E: Are able to swallow capsules or tablets Exclusion Criteria: * Have received more than 2 previous lines of cytotoxic chemotherapy (if receiving cisplatin, 5-FU or pemetrexed) * Must not have taken an unapproved drug as treatment for any indication within the last 28 days prior to starting study treatment * Must not have an active symptomatic fungal, bacterial or viral infection, including human immunodeficiency virus (HIV) or Hepatitis A, B, or C * Must not have a serious heart condition, such as congestive heart failure, unstable angina pectoris, or heart attack within the last three months * Must not have a family history of long QTc syndrome * Must not have a serotonin-secreting carcinoid tumor or a prior history of drug-induced serotonin syndrome * Must not have acute leukemia * Part E: Have insulin-dependent (type I) diabetes or a history of gestational diabetes * Part E: Prior treatment with a PI3K/mTOR inhibitor