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The best undrafted free agent in NFL history is quarterback Kurt Warner, who helped architect the St. Louis Rams' "Greatest Show on Turf" from 1999-2001, and then later returned to the Super Bowl at the helm of the Arizona Cardinals offense in 2009.
Warner was initially a 1994 undrafted free agent signing of the Green Bay Packers, but he was released prior to the regular season. Instead, he opted to join the Arena Football League's Iowa Barnstormers, where he had great success.
In 1998, the St. Louis Rams signed Warner, and he spent time in the NFL's European League before coming back to the Rams to be their third-string quarterback. Injury forced Warner into the starting spot in 1999, which led to the Rams reaching and winning the Super Bowl, where Warner was named the game's MVP. The Rams returned to the Super Bowl in 2001, but lost to the New England Patriots.
Warner spent 2004 as the New York Giants' starting quarterback and then joined the Arizona Cardinals in 2005. The Cardinals reached the Super Bowl with Warner as their starter in 2008, ultimately losing to the Pittsburgh Steelers.
In his NFL career, Warner has completed 2,666 of his 4,070 pass attempts for 32,344 yards, 208 touchdowns and 128 interceptions. He's a four-time Pro Bowler, was named an All-Pro twice, was NFL MVP twice and Super Bowl MVP once. He also holds the top three Super Bowl passing records.
(Unless otherwise noted, all information sourced from Pro Football Reference and Wikipedia)