Which games have ever been rated #1 on Boardgamegeek.com's Geek Rating?
Strangely, Wikipedia had this information:


  
  Paths of Glory (August 19, 2001 - February 20,2002)
  Tigris and Euphrates (February 20, 2002 - 2002)
  Puerto Rico (2002 - August 2008)
  Agricola (August 2008 - December 2010)
  Twilight Struggle (January 2011 - December 2015)
  Pandemic Legacy: Season 1 (January 2016 - Present).
  


As Jefromi's answer shows, the dates aren't quite as nicely laid out as Wikipedia shows, but these are the games that have been #1 at some point in time based on persistent user ratings.

If we expanded it to games that rose to #1 because of bugs or user coordination, three more games are added: (1) The Game of Life, (2) Monkey Auto Races, and (3) Die Macher. The specific reasons for each are described on the Wikipedia page.

The games that have hit #1 are also listed on BGG in one of the Completely Useless Statistics threads.



Update:

As of December 29th, 2017, Gloomhaven is the #1 rated board game.