Greer is an unincorporated community and census-designated place in Apache County, Arizona, United States. It lies at an elevation of approximately 8,300 feet (2,500 m) in the White Mountains of Arizona, and is surrounded by the Apache-Sitgreaves National Forest. As of the 2010 census it had a population of 41. Greer was founded around 1879 by Mormon settlers from Utah. The Greer post office has the ZIP code of 85927.