Our hero certainly spends a lot of time rescuing that princess, so he must really care about her. And it's a safe bet that she really cares about him, so it's only natural to assume they're more than Just Friends, right?
But wait...had anyone actually said that the two of them are together, or are you just assuming? Is the character denying the truth or stating it when he says "She Is Not My Girlfriend"? Is it Not What It Looks Like — or is it? We could ask questions all day.
Either way, in a serial work, this allows the author to gently cover their bums when it comes to either Ship Sinking ("Hey, the ship never set sail in the first place") or Cleaning Up Romantic Loose Ends ("The potential was there from the start"). Expect there to be some fans who pair the couple up regardless - and the writer may state that they Hooked Up Afterwards, or leave it to the reader's imagination. Naturally, this goes hand-in-hand* Though whether palms cupped or fingers laced, we couldn't tell you with Ship Tease.
If the relationship is left ambiguous because the characters are of the same sex, it's Hide Your Lesbians.
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Anime & Manga
InuYasha: Sesshoumaru and Kagura. Rin's observations and Kagura's thoughts imply that Kagura's in love with Sesshoumaru. Moryoumaru's comments and Totosai's observations confirm that Sesshoumaru's got feelings for Kagura but Takahashi leaves the nature of their relationship open to interpretation.
Tower of God: Baam chooses to follow Rachel where ever she may go, even if that includes fighting sea monsters even though he had to be taught why people would use violence at all few months prior. Many note that this has a romantic notion to it and the author (of course) hasn't said a thing about their relationship status. She could be a mother figure, big sister, best friend or love interest.
Natsu/Lisanna, though Lucy throws this one into rather gray territory, leaving Natsu with two implied love interests and nothing solid.
Lyon/Sherry, though this was later busted for Lyon falling for Juvia and Sherry and Ren.
Lucy has multiple potential love interests, the most prominent being Natsu and Loke.
Gajeel and Levi, especially in later arcs. Almost constantly together, invested in each other's well being, etc., yet it still has not been clarified if they're a Beta Couple or Platonic Life Partners.
Mazinger Z: Kouji and Sayaka were the Trope Codifier for Belligerent Sexual Tension, spending most of the time together and were very close at each other, and after the Final Battle they traveled to America together, but they never became a couple. The anime makes Sayaka's feelings very explicit, but downplays Kouji's ones and showing him getting pervy about other women, but both manga versions (especially Go Nagai's original story) are more explicit about them wanting to pursue a relationship. They never quite reach the Official Couple status, but most fans take that status for granted. In other continuities they did become Official Couple on panel, though.
Great Mazinger: Tetsuya and Jun did not get together on panel, but there were tons of Ship Tease throughout the series. Jun spent the whole series chasing Tetsuya as he was Oblivious to Love, and he got jealousy when Jun got some suitor or he thought she would leave him.
UFO Robo Grendizer: Duke and Hikaru clearly cared for each other and loved spending time together, but it was never explicitly stated they were together.
In several Pokémon adaptations Misty was this for Red/Ash. It's a point of controversy in the anime: the original writers stated that the two are too young to recognise their feelings, and most of the dialogue implying attraction in the Japanese version leaned more towards Platonic Life Partners than romance. The English dub just bluntly promoted her to All Love Is Unrequited - many instances of dialogue were rewritten in an attempt to push the couple, and even more notoriously, Misty's farewell sequence had the scenes depicted in the flashback completely changed (the original focused less on her as a Love Interest, and more on her as a trainer and as a friend).
James and Jesse too, in the anime. There have actually been a few subtle hints that James does have feelings for Jesse, but never anything serious, but fans tend to think otherwise.
Axis Powers Hetalia is a confusing case. While there are undoubtedly several potential pairings who get enough Ship Tease to justify calling it this, some characters are involved in more than one of these pairings, which leaves a situation where a character can have numerous other characters where a hook-up would have received enough subtext to not come as a surprise.
While possibly not the case with their counterparts in Go Lion, Keith and Princess Allura in Voltron are widely considered the show's Official Couple, but their feelings for each other are never stated to go beyond friendship. It's possible that they are in love with each other and just don't realize it, but aside from a ton of Ship Tease the show doesn't really portray them as a couple.
Suzuna and Sena of Eyeshield 21. The two have a meet cute introduction to each other, people often think they're a couple, and Suzuna states that she doesn't view Sena as a little brother (unlike Mamori). Sena's feelings for Suzuna are left rather ambiguous as well, other then the fact that he's rather protective of her.
In Thieves Like Us, the exact relationship between Con and Motti is difficult to pinpoint. While the two mostly interact as friends and co-workers, there is one part in the first book where Motti looks after Con while she has a Heroic BSOD and admits he feels something for her, but suspects she wouldn't reciprocate. When she wakes up, she tells him she heard some of what was going on around her, and adds that he is the sort of guy she could be with. Even after this though, the two never actually share romantic scenes and show no indication that they might do so.
Generally assumed for Ian Chesterton and Barbara Wright, as well as Ben Jackson and Polly. Ian and Barbara were eventually made canon by a throwaway line in The Sarah Jane Adventures, which revealed that they had married.
Despite being No Hugging, No Kissing the original series has several of the Doctor/companion relationships possessing a romantic tinge, namely Three/Jo, Four/Sarah Jane, and especially Four/Romana. Four/Sarah Jane was finally made canon after thirty years in the New-Who episode "School Reunion", in which the Tenth Doctor told Sarah Jane that his Fourth incarnation had loved her but chosen not to pursue the relationship for various reasons.
The Doctor and the Master are widely assumed to have either a highly dysfunctional relationship or a romance that went sour. Not at all hurt by the new series laying on Ship Tease so thick it could be cut with a chainsaw.
The Second Doctor and Jamie, mostly because the actors both thought it would be fun to play them like this. Keep in mind that this was in the 1960s, when homosexuality was still illegal in Britain (although by that point barely ever prosecuted).
Rose's relationship with the lonely Dalek in the episode "Dalek". Both of them seem to fall in love with each other a bit, the Dalek agonising over the positive emotions it feels towards her and Rose herself being awed by the Dalek's vulnerability, monstrous beauty and power, but it's left implicit because having either character acknowledge it for the benefit of the audience would turn it from being heartwrenching into pure Squick due to the nature and personality of the Dalek species, as well as create a case of Never Live It Down for the reboot Daleks.
In an alternate future of Star Trek: Enterprise, the now elderly Captain Jonathan Archer is being cared for by T'Pol. However, an injury prevents him from remembering all that's happened since he was last on the Enterprise. T'Pol says to him that she fully understands that he might consider all this to be an illusion or an elaborate deception. To alleviate his fears she tells him the story of an old girlfriend he wanted to marry back on Earth. The stunned Archer wants to know just what kind of 'relationship' he and T'Pol have that he'd ever tell her the story. She'll only say their relationship has "evolved".
Blake's 7 did this to various degrees for Blake/Jenna, Avon/Cally, Tarrant/Dayna, and Avon/Soolin.
Jim Stafford's song "My Girl Bill", despite its name, uses Exact Words to imply that the (male) singer is in a relationship with a man named Bill, who he is calling "My Girl Bill." At the end, it turns out that he and Bill are competing over the same woman; the singer has been saying "Don't get between me and my girl, Bill."
The best example would have to be Princess Peach from Super Mario Bros., and it may extend to Princess Daisy from Super Mario Land. No romantic connection was made explicit, yet everyone assumes.
Lampshaded by Luvbi in Super Paper Mario, and it's shown that the other characters seem to be just as clueless about whatever's going on between Mario and Peach, which leads Luvbi to think it is an unrequited crush.
The Paper Mario game before that seems to have NPCs throw out plenty of Ship Teases, and the things Peach says while in captivity seem to imply that she thinks of Mario as a lot more than a friend, however the interaction with Luvbi is probably the only time where Mario and Peach's relationship is explored beyond the two knowing each other, unless you count Bowser accusing Mario of "trying to get between he and Peach".
The various games and other sources seem to go back and forth with how "official" Mario/Peach is. Some outright confirm their relationship, others only imply it.
Link can get multiple Implied Love Interests in some games, which helps add to the ambiguity. The Hero of Time had Saria and Malon in addition to Zelda as implies love interests, not to mention the one-sided (at the least) Interspecies Romance crush that Ruto had on Link. Twilight Princess Link had Ilia as a more overt love interest, and Midna and (sort of) Zelda as possible implied ones.
Disgaea has a few of these, the most telling is the ambiguous relationship between Laharl and Flonne. While Laharl repeatedly claims to not like her, his actions in the game suggest otherwise, going as far as to attempt a Heroic Sacrifice. At the time, she was Not Quite Dead in the canon ending while in the normal ending he really does go through with it.
Persona 3 is quite heavy with two of the Love Interests, Yukari and Aigis. Yukari gets quite a few scenes with the protagonist in the beginning of the plot, though it dies down eventually. Aigis is a full-blown Stalker with a Crush, justified that she sealed Death into the protagonist but doesn't remember. It should also be noted that the protagonist dies in Aigis' arms in all versions of the game with the exception of New Game Plus in the PSP version. In both cases, completing the girl's Social Link makes her relationship with the protagonist overtly romantic, but since that's also the case with every other female Social Link in the game, none of whom are mutually exclusive, the issue of whether there can be said to be a canon Love Interest and, if so, who it is continues to be a subject of debate in the fandom, with Yukari and Aigis as the two most likely contenders.
It's worth noting that in "The Answer", Aigis and Yukari are the two party members who take the protagonist's Heroic Sacrifice especially hard.
Done very well in Cave Story, where the protagonist and his female counterpart Curly Brace are almost never seen in particularly romantic situations, but it's still clear that they care about each other in some way.
Liara T'Soni in Mass Effect. In Mass Effect 1 she was the only character who could be romanced by both genders, and by later games she gets upgraded to being "Shepard's closest companion" regardless of how you actually treated her in conversations. Most damning enough is that, before the extended cut of Mass Effect 3, she was always the player's final thoughts as Shepard dies unless they romanced Ashley or Kaidan. Regardless, it's a contentious subject amongst the Bioware fanbase.
Subverted with Princess Elincia for Ike in Path of Radiance. There seems to be a teeny bit of ship tease between the two (mostly in the American localization) but in the end they remain platonic friends and in the sequel, Elincia is more or less canon with her loyal knight Geoffrey.
The Sonic the Hedgehog series generally avoids the subject of romance, aside from Amy's unrequited crush on Sonic, but there have been implications between Knuckles the Echidna and Rouge the Bat, and Sonic and Blaze the Cat.
Ribbon from Kirby 64: The Crystal Shards is the closest thing to a love interest in the Kirby series. If you get the good ending, which requires 100% Completion, Ribbon will give Kirby a kiss, making him blush. It depends on whether you interpret the kiss as a romantic one or just a simple "Thank you" kiss. This is probably why Ribbon is so popular despite only being in one game.
Cloud and Aerith from Final Fantasy VII. Although it is implied that Aerith was attracted to similarities to her previous boyfriend Zack and Word of God confirms Tifa is the official love interest, there's still massive amounts of subtext. The Kingdom Hearts games play into this by presenting Aerith as Cloud's implied love interest. It's also worth noting that in the romantic mini-gamenote That determines which of the playable females (or Barrett) will go on a date with Cloud in the Gold Saucer, Aerith is the easiest one to get. In order to get Tifa, you must first alienate Aerith. Then Aerith dies, making a relationship a tad tricky.
Advent Children plays into this as well. Tifa and Cloud are living together but display no couple-type habits and their scenes together could easily be interpreted as a platonic bond. Then there's that flirty smile she gives him at the end. It also doesn't help that the movie presents Aerith as The Lost Lenore.
Final Fantasy X does this with Lulu and Wakka. She previously dated his brother Chappu who died in battle. Tidus simply suggests to her that she get with Wakka instead at one point. She does share a tender moment with him in the middle of the game but it's left vague. The sequel is two years later and they're suddenly married with a baby on the way.
The sequel also gives Rikku one as well. A lot of the scenes with Gippal are optional and if you miss some of them, it's this trope. If you get all of them (for 100% Completion) then it results in a Maybe Ever After.
All of the player characters in Final Fantasy XII. Vaan and Penelo have grown up together and they're certainly quite close. They are implied to be together by the end of the game. Balthier and Fran have also been partners (as in sky pirates) for years and there are obvious signs of flirting - as well as a handful of tender moments between them. Ashe and Basch do have an interesting relationship and this line from Penelo "I know Ashe misses Basch" implies some kind of romantic connection.
Rai/Kim from Xiaolin Showdown is a very popular ship on FF.net, and while there are indications that they might like each other in the series (Rai dreaming about being on a date with Kim, Rai being the only one of the guys that Kim kisses on the cheek, the third season episode "Dreamstalker" confirming Rai's side), it is never outright stated whether Kimiko reciprocates or not.
Deconstructed with Finn and Princess Bubblegum. Sure, Finn has a Precocious Crush on PB and he frequently rescues her, but she only thinks of him as a close friend at most. They come very close to confirming their ship when she is de-aged, but it's thoroughly nixed when she returns to her original age. One episode involved him singing a song about how confusing it was ("What am I to you? Am I a joke, your knight, or your brother?") that also includes his sentiment that it doesn't really matter as long as they are at least friends and get to hang out. He eventually gets over the romantic aspects when he meets Flame Princess.
Aerrow and Piper on Storm Hawks are close friends and have several intimate moments throughout the series. Their relationship never goes into anything explicitly romantic, but you could hardly be faulted for assuming it was intended to be. Then again, they could just be Platonic Life Partners.
Fred and Daphne in the original Scooby-Doo, Where Are You! cartoon may very well be the Western Animation Ur Example. In the original series, none of the character were romantically linked. The closest the series came to outright Ship Tease was occasionally having the pair dance together. This was enough for the general audience to assume they were couple. It's become so synonymous with the franchise that manymanyfutureadaptations have took the subtext and just ran with it.
Mulan and Captain Li Shang in Mulan. They blush around each other and Mulan asks the captain to stay for dinner, but no one outright says that they are a couple. Subverted in the 2nd Mulan movie when they get married.