Twilight Struggle
"Twilight Struggle" was the highest-ranked game on BoardGameGeek from December 2010 to January 2016. As of August 2018, the board game occupies a fifth spot. Being released in 2005 it is the oldest board game in the overall top 10 ranking.
Backgammon
Race board games involving dice have existed for millennia in the Near East and eastern Mediterranean, including the game senet of Ancient Egypt. The ancient Egyptian game senet was excavated, along with illustrations, from Egyptian royal tombs dating to 3500 BC. Though using a board that is quite different from backgammon, it may be a predecessor.