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 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.