(covers information from several alternate timelines)
"Space, the final frontier. These are the voyages of the starship Enterprise. Its five-year mission: to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no man has gone before."
- - James T. Kirk, opening narration to Star Trek: The Original Series.
In 2257, Captain Christopher Pike mentioned that he had been on a five-year mission aboard the USS Enterprise, a considerable distance away. They had not been ordered to return during the Federation-Klingon War, which had taken a toll on the crew. (DIS: "Brother")
One of the greatest chapters in Starfleet history was another five-year mission of the USS Enterprise from 2265–2270 under Captain James T. Kirk. (Star Trek: The Original Series, Star Trek: The Animated Series, VOY: "Q2")
On May 12, 2260 of the alternate reality, the USS Enterprise, under the command of Captain James T. Kirk, began the first official five-year mission of deep space exploration into uncharted space that had ever been attempted by any starship. By the third year, the ship had been in uncharted space for an extended period. This had stretched the ship's mechanical capacities, although the engineering department was up to the job. The rest of the crew likewise continued to act admirably despite the rigors. By day 966, the crew were looking forward to a resupply stop at Starbase Yorktown. From there, the ship continued on a rescue mission to Altamid, where it was destroyed. The crew later continued their mission in the USS Enterprise-A after it was finished. (Star Trek Into Darkness; Star Trek Beyond)
The 2265-2270 five-year mission of the USS Enterprise mission was chronicled in Star Trek: The Original Series and Star Trek: The Animated Series, based on the opening narration to those series. The Star Trek Encyclopedia and Star Trek Chronology have conjectured that Captain Robert April had one five-year mission and Captain Christopher Pike had two five-year missions with the Enterprise before Kirk assumed command. These sources also state that Kirk had a second five-year mission after Star Trek: The Motion Picture. DIS: "Brother" later made canon that Pike did indeed have at least one five-year mission as commander of the Enterprise which, based upon dates given in the series, would have been underway at the time of "The Cage".
Until 2260 in the alternate reality, a five-year mission of deep space exploration was unprecedented. Because the alternate reality was a divergent timeline that split from the original universe in 2233, there presumably had not been any official five-year missions prior to, at least, that year in the prime reality.
No references have been made to five-year missions after the 23rd century. Instead, the opening narration of Star Trek: The Next Generation refers to the Enterprise's "continuing mission". However, in 2363, the USS Olympia embarked on an eight-year mission. In an interview in the DS9 Season 3 DVD, Ira Steven Behr said that the USS Defiant was "on a five-year mission to kick ass."