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Greer is an unincorporated community and census-designated place in Apache County, Arizona, United States. Located within the White Mountains of Arizona and surrounded by the Apache-Sitgreaves National Forest, Greer is the highest town in the state at an elevation of approximately 8,400 feet. As of the 2020 census it had a population of 58. Greer was founded circa 1879 by Mormon settlers from Utah. The Greer post office has the ZIP code of 85927.