What medical schools do not require the MCAT?
For schools within the United States, only Dartmouth does not require the MCAT. That is a theoretical lack, however; in practice, every U.S. medical school requires it.
Some offshore schools, such as St. James in Anguilla, do not require the MCAT. French universities also do not require the MCAT (University of Montreal is an example), because the exam is not offered in French. The University of Ottawa also does not require the MCAT, because it is a bilingual university.
Two other methods of skipping the MCAT exist. If you are in high school or have not earned your baccalaureate, you may apply to one of several highly competitive college programs that combine undergraduate study with medical school (sequentially, that is). No MCAT is required but substantial achievement must be demonstrated to be accepted to one of these programs.
Second, at least one school offers a pre-med program that guarantees entry into the medical school sans MCAT (on successful completion of the requirements, one would imagine). International American University in St. Lucia is an example.