The Tarrasque debuted in MMII, 1983, to provide a near-indestructible monster
The Tarrasque appeared first in the print in the Monster Manual 2 for AD&D 1st Edition in 1983, which was entirely credited to Gary Gygax as the author. It has since appeared in the Monster Manual for every edition of the game.
Before then, it however was played in the extremely high level home campaign of Francois-Marcela Froideval, who also is the Author of Chronicles of the Black Moon, and was an early D&D player and friend of Gary Gygax. Gary occasionally played in his campaign with his own character, Mordenkainen:
At 14th level through the next few above that he played for a good bit of time in Francois Marcela Froideval's campaign. At those levels Mordenkainen was a low-level "flunkie" type, as the movers and shakers in that setting were of high 30th and 40th level. Francois had a complete campaign based on ultra-high level characters, and believe me it was filled with challenges and a very real sense of danger for PCs of under 30th level, I should think. there was a lot of roleplay, and the wrong dialog coming from a chatacter could be fatal... [Gary Gygax, ENWorld Forum, Post 1975]
Because the campaign was so high level, Francois needed monsters with extreme statistics. He created the original Tarrasque, losely based on the Tarrasque of French mythology (he himself was born in France):
The tarrasque is a fablous monster from French legend, and it was French author Francois Marcela Froideval who called the beast to my attention, did up the stats for it. I thought the game needed at least one near-undestructable creature from myth, so it made the book [Gary Gygax, ENWorld Forums, Post #1456]
The Tarrasque thus was included in the rules, with the design intent to provide a near-indestructible monster for the game.
"Ay pronunseeAYshun gyd" by Frank Mentzer in Dragon #93 (1985) gives the pronounciation as:
Among others, there is an adventure set in the Forgotten Realms "How the Mighty are Fallen" (for 2nd Edition AD&D) that features the Tarrasque.
Ed Greenwood and Johnathan M. Richards provide an in-game “The Ecology of the Tarrasque” in Dragon Magazine #359, pp. 88–95 that intersperses rules mechanics, and provides a Tarrasque advanced to CR 30 for the 3rd Edition ruleset.
"History Check: The Tarrasque", by Jeff LaSala in Dragon Magazine #418 is an adventure for 4th Edition.