While Princess Zelda is a key character in much of The Legend of Zelda, there are some games in the series that don't actually have Zelda in them at all. Nearly every version of Princess Zelda has supported Link in the games to some degree, from sporadic moments of cooperation to continuous aid. However, not every one of Link's adventures has contained the titular princess.
Some Legend of Zelda games only show the princess briefly or feature Zelda in an alternate form or disguise. Zelda appeared in The Legend of Zelda: Majora's Mask, though only in a scene where Link recalls a memory of her. Similarly, the pirate Tetra from The Legend of Zelda: Wind Waker is technically neither Zelda nor a princess, yet she is referred to as "Princess Zelda" due to descending from Hyrule's royal family and possessing the Triforce of Wisdom. Still, there are some series entries where Zelda is entirely absent.
Very few Legend of Zelda games actually exclude Princess Zelda. However, the titles in which she is absent often lead to more emphasis being placed on a new character, or on Link himself. Here are the handful of games in which Zelda is doesn't get any screen time at all.
Zelda Is Only Mentioned in The Legend of Zelda: Link's Awakening
Princess Zelda does not appear in The Legend of Zelda: Link's Awakening, though she is mentioned once. Given that Nintendo developers saw Link's Awakening as a Zelda parody, the princess' absence is just one of many unique decisions made in the game. Instead of Zelda, Link's journey focuses heavily on a girl named Marin, a denizen of Koholint Island who is beloved by all. When Link washes up on the shores of Koholint Island in the game's opening, it is Marin who discovers him, and she continuously provides Link aid in place of Zelda. Interestingly, when Link initially awakens at the beginning of the game, he mistakes Marin for Zelda, to which she expresses confusion and corrects him. Besides this one name-drop, Zelda is otherwise absent for the entirety of the original Link's Awakening and its 2019 remake.
Only Zelda's Dress Appears in The Legend of Zelda: Tri Force Heroes
While Princess Zelda does not appear in the multiplayer game The Legend of Zelda: Tri Force Heroes, Link can wear her dress. Tri Force Heroes takes place in the kingdom of Hytopia rather than the familiar setting of Hyrule. To save Hytopia's Princess Styla from a curse, Link cooperates with two teammates (controlled one at a time or by other players) to defeat a witch called The Lady and is armed with a number of outfits providing special benefits. One of Link's possible outfits, the Legendary Dress, directly references Zelda's dress from The Legend of Zelda: A Link Between Worlds.
Zelda Only Appears With a Secret Code In The Legend of Zelda: Oracle of Ages and Seasons
Princess Zelda technically does appear in both The Legend of Zelda: Oracle of Ages and Oracle of Seasons, but only after inputting a secret code. Ages and Seasons take place in the lands of Labrynna and Holodrum respectively, and Link must defeat the evil forces sending them into chaos. In a normal playthrough of either game, Zelda is mentioned by name, but never appears. However, completing one of the games will award the player a secret code which, when inputted at the start of the other, begins a Linked Game. While the games are normally two separate scenarios, a Linked Game recognizes one as a continuation of the other, resulting in a greatly modified story. It is only in these revised plotlines that Princess Zelda will appear in the Oracle games, and her presence drastically changes the final acts of each game.
The outright exclusion of Princess Zelda from future Zelda games seems unlikely given her importance. Some speculate that there will be a playable Zelda in Breath of the Wild 2, which would put the princess in the most prominent role of any incarnation of her. Imagining Zelda in such an important position makes it even harder to fathom how certain The Legend of Zelda games did not have Zelda in them at all.