Putnam County, MO has been consistently Republican since at least 1932. I'm not able to find county-level results from before then. Maybe someone here can? It's definitely the longest one-party streak of any Missouri county, but I have no idea of exactly how long it is.
Actually, Gasconade County is the longest one-party streak of any County, not just in Missouri, but nationally too. Both Gasconade, and Putnam though, along with Ozark and Taney Counties have voted Republican every time since after the Civil War ended though.