Republican Jan Brewer, after being sworn in as the twenty-second governor of Arizona, addresses those gathered during inauguration ceremonies on Jan. 21, 2009, in Phoenix. Ross D. Franklin/AP hide caption toggle caption Ross D. Franklin/AP The Two-Way Arizona Gov. Jan Brewer Will Not Seek Another Term March 12, 2014 To run for another term, it would have required a legal battle to challenge the state's term limits. Brewer completed the final year of Janet Napolitano's term. She won another four-year term in 2010.