The first example of a recurring Link is obviously in the games Zelda and Zelda II. I’ve named him the Hero of Decline as he appears at the end of the Decline branch of the timeline, according to Hyrule Historia (it’s an exercise to the reader which of Spirit Tracks, Four Swords+ and Zelda II is the latest chronologically). Before the concept of reincarnation, iterations, and cycles, the second game was explicitly a straight sequel to the first with the same characters (except Zelda, strangely. She was replaced by a comatose ancestor with the same name- I suppose the first example of iteration). Link has aged three years by the time of the sequel.
This Link also is probably the one in the Legend of Zelda Game Watch from Nelsonic, and the Zelda Game & Watch from Nintendo. My theories: the Game & Watch is set between the two main games, as the Link in artwork is squat, resembling the first one. Some dragons kidnap Zelda and rehashing the first game, Link has to reassemble the Triforce of Wisdom. The lankier, older Link from Zelda II is reproduced in the artwork of the Game Watch, on the other hand. It takes place in a single cave with Link assembling another Triforce. This time my theory is that this Triforce is the one of courage, this cave being a test of Link’s courage for him to obtain the physical Triforce as seen later in Wind Waker, him having proven his right to it by defeating his shadow at the end of Zelda II. So it goes Zelda 1 -> Game & Watch -> Zelda 2 -> Game Watch.
There’s a different kind of iteration on this Link—adaptation iteration. The Valiant comics and the cartoon series were based on the first two games and set after it, in what you might call another alternate branch of the timeline in which things get a whole lot cheesier and surreal. You see, instead of the route some adaptations take in adapting game logic to more realistic alternatives, these media made their world follow some bizarre game logic in a superficially realistic setting. They were also strangely non-violent, with Link “zapping” Ganon’s minions with his “Crissword”, magically sending them to his “Evil Jar”. And there was all the cheesy, mugging dialogue. Ugh. This means that while implicitly based on the same Link, the version seen in these sources is so different in terms of personality we have to separate him for the sake of our sanity. (Well excuuuuuuse me, Princess!)
A further adaptation removed another step was the Link seen in the Captain N cartoon, which ran at the same time as the Zelda cartoon. In some ways he’s a continuation, with a similar personality but a more mature look and attitude (making this perhaps the latest pseudo-canon source in its own timeline). However, this cartoon is explicitly set in a weird “videogame” crossover universe where everyone knows they’re game characters or something. So I don’t really know what to make of it.
Of course, the last two points are not so relevant, until we consider the CD-i games. Oh yes. The first two at least, Link: The Faces of Evil and Zelda: The Wand of Gamelon, base their character depictions on the cartoon, with designs more from the first two games. King Harkinian of Hyrule here has colour in his hair, so they must be set before the cartoon and comic in this strange timeline branch, where his hair has gone white. Now you see what I mean by Decline!
Zelda’s Adventure, the third CD-i game (different developer this time) is harder to place and has a very different tone, especially in the cutscenes. They’re live-action and less campy but still hammy, if that makes sense. Let’s say for simplicity it occurs after the other two CD-i games, in development order, and that the universe fell down a realism hole briefly before going a bit wacky again in the cartoon.

The Hero of Decline, now that I look at him, seems to have had a lot of appearances. And I didn’t even mention the two Nintendo Adventure Books, which depict original plots: The Crystal Trap and The Shadow Prince. This Link also makes cameo appearances, like several other Nintendo characters, in F-1 Race and Tetris on the Game Boy and NES, respectively, in the context of those games with a new sprite (an illustration of the concept of the Videogame Crossover Universe, depicted differently in Captain N, Smash Bros., etc—you can tell it’s Decliney due to the crucifix shield and flute, exclusive to the first two games). Other games, such as the WarioWare series, Tetris DS, and Picopict, present Link in his sprite art appearance from Zelda 1, in an appropriate context (so they don’t really count, as he’s still in Hyrule during those events that are merely being depicted).
Well, that’s the original Link. The comics claim he is a traveller from the neighbouring kingdom of Calatia, but either way, by all accounts he is an unassuming youth who takes on the mantle of destiny and proves himself worthy to the goddesses to possess the Triforce of Courage. Of course, we find out later that he’s one of the last Links in a very long line of heroes who is retrospectively revealed to be following traditions of previous heroes. We leave him still defending a Hyrule ruined after many years of decline. Can he and Zelda restore the kingdom to its previous prosperity?