The warp 10 barrier is a theoretical barrier for a starship with warp drive. Warp 10 is regarded as infinite velocity, so theoretically any vessel travelling at warp 10 would exist at all points in the universe at once.
In 2267, the USS Enterprise exceeded warp 10 during its encounter with the Nomad probe. Upon hearing that the ship with traveling at over warp 10, Montgomery Scott stated that it was impossible. (TOS episode: "The Changeling")
The Daystrom Institute's transwarp project was designed to break the warp ten barrier. Torias Dax was critically injured during a test of the technology aboard the shuttlecraft Infinity. (DS9 - The Lives of Dax short story: "Infinity")
In 2364, the USS Enterprise-D travelled at speeds nearing warp 10 when it was caught in a gravimetric wave, it was only able to break free be using the slingshot effect. (TNG - The Space Between comic: "History Lesson"). It also appeared to travel in excess of Warp 10 during the Kosinski experiments (TNG - "Where No One Has Gone Before")
In 2372, the warp 10 barrier was broken by Lieutenant Tom Paris using the shuttlecraft Cochrane and a rare form of dilithium that was discovered in the Delta Quadrant by the USS Voyager. While Paris successfully broke the barrier, it was later discovered that the effect caused hyper-evolution in the humanoid body, and later tests were abandoned. (VOY episode: "Threshold")
In the Kelvin timeline created by Nero, Section 31 agent, Lieutenant John Harrison was able to revamp the USS Vengeance's warp drive to give it unprecedented capability. Lieutenant Yuki Sulu estimated that Harrison's work made it possible for Vengeance to get close to warp 10. (TOS - Khan comic: "Issue 4")
The Enterprise's rate of speed on stardate 5630.7, estimated at the "old scale" of warp factor eleven, is approximately warp 8 by 24th century standards. (TOS episode: "Is There in Truth No Beauty?" , reference : Star Trek Encyclopedia (W section))