Jean Kirstein is the cynical Jerkass to Eren's idealistic hero, in Attack on Titan. The two immediately come into conflict due to their differing ideas, and nearly end up putting things behind them. That he ends up falling for Eren's beautiful, violently-protective foster sister results in him becoming a Green-Eyed Monster and adds further fuel to their rivalry. However, Jean ends up accurately predicting that Eren's lack of forethought would come back to haunt him and proves to be the better soldier of the two. It turns the whole trope on its head, however, in that Jean doesn't end up coming over to Eren's side because of anything the later did, but rather through Jean's own personaltragedies and development into a promising young leader.
Bleach: Ichigo's had several over the years, but his most notable rival is arguably Grimmjow (who serves as his Evil Counterpart), whose hatred of him borders on obsession. What started as an attempt to outdo Ulquiorra eventually lead to arguably the most intense grudge between two people in the entire series. The animosity between Grimmjow and Ichigo finally reached a boiling point in the Hueco Mundo arc, where the two finally had an uninterrupted fight in which Ichigo won. Grimmjow hasn't been seen since though he has been confirmed for the final, manga-only arc.
A little more complicated than that. Suzaku plays friendly enemies with Lelouch, due to their friendship but drastically opposed viewpoints. Though Lelouch is initially not aware, and feels quite betrayed when he finds out, he sort of understands Suzaku's views and situation, going so far as to save him twice, despite it. Up until the second season, where things get... generally more violent, and their respective actions, much less forgivable.
On the other hand, Kallen and Suzaku more easily fit the mould of the rival, with both being exceptional pilots, and the same ultimate goal.
Death Note: Light and L are classic examples, foils with similar traits, differing ideals and near-opposite appearances to offset their near-identical approaches to problems. They suffer from a serious case of Rivals Team Up under odd circumstances as well, though it both was and wasn't true teamwork.
Half the cast of Dragon Ball Z, Vegeta being the most prominent example, with Yamcha being the first.
Hakuoh and Kokujo from Duel Masters, the former doing a total 180 in every aspect after Shobu beats him, while the latter never really stops being the rival, softening only the slightest bit.
Gajeel and Erza from Fairy Tail could both be considered this to Natsu.
Gray was as extreme an example of a rival to Natsu as it could be in the early Fairy Tail days, but Gajeel seems to have taken his role starting with the Fighting Festival arc.
Kenshiro had Shin and later Raoh as a primary rival in Fist of the North Star. His friendly rival/brother-in-kung-fu Rei had Juda as his.
Rin considers Haruka his rival in Free!. It's a little one-sided, as Haru doesn't like to compete and doesn't seem especially interested in beating Rin. However, when they were children, Haru was miffed that Rin seemed to "understand the water" more than he did.
It is commented on later that Haru is actually more fired up when it comes to Rin, and by the time Rin and Haru reconcile, they've gone on to heatedly competing over silly competitions like catching crabs and who gets the better fortune.
Although this case in interesting in that neither of them know anything about each other nor each others names. They meet once but say very little and don't know who the other is. In fact once Shinn upgrades to Destiny Kira doesn't even know he's the same pilot he's fought before. So they are effectively faceless enemies to each other for their entire rivalry.
Athrun Zala is Kira's rival in the orginal Gundam SEED and probably his main rival overall. This is a unique example for the Gundam franchise in that they know a lot about one another. They were childhood friends and often find it difficult to fight one another. The show milks this dynamic for all it's worth, having Athrun attempt to capture Kira alive and having oodles of fun when messing around with the dynamic the two boys' love interests have.
In Gundam 00 Graham Aker is the rival to Setsuna F. Seiei. Interesting in that Graham is at first a typical mecha pilot, but it's in season two, after we've met him, that he becomes the traditional Char Clone. When the movie comes around Graham has come to understand Setsuna's perspective and prays for his recovery so they can both fight the oncoming alien threat.
Jerid Messa is Kamille Bidans's rival in Mobile Suit Zeta Gundam. There rivalry is definitely one fueled by revenge, and is actually more of a mutual hatred than just a rivalry, considering they both have taken the lives of many important people in the other's life.
Anavel Gato and Kou Uraki become rivals in Mobile Suit Gundam 0083: Stardust Memory. At first Gato considers Kou as nothing more than a rookie that's in over his head, but soon enough he recognizes him as a worthy foe after their duel and the destruction of the GP-02A and the GP-01Fb
Alucard and Alexander Anderson are this in Hellsing. Due to Anderson's humanity, regenerator abilities, and ridiculous levels of determination, Alucard views him as a Worthy Opponent and relishes every fight they're able to engage in, sometimes even overlooking his primary mission in order to fight him. And although Alucard visibly breaks when Anderson forsakes his humanity in a bid to defeat him, he still respects the priest in the end and understands why Anderson made the choice that he did.
Akira in Hikaru no Go: The series premise makes turning evil and/or the need for a rescue, embarassing or otherwise, highly unlikely. Mass quantities of Foe Yay are used, seemingly to make up for it.
Immortal Grand Prix: When Takeshi Jin was hired by Team Satomi, incumbent pilot Riven felt put out by the decision, and the two were bitterly at odds until Riven finally decided to quit and join rival Team Sledgemama. As it turned out, Takeshi connected with Riven much better as a rival, and they've since become much friendlier.
Takahashi Keisuke (specifically, because he best fits the "intolerable ass" requirement), and at least half the cast of Initial D.
Magical Girl Lyrical Nanoha A's had one Wolkenritter to match up against each of the main combatants on the cast; Signum vs. Fate, Vita vs. Nanoha, and Zafira vs. Arf. Signum and Zafira were more like Worthy Opponents to their respective counterparts, but Vita plays the trope straight.
Magical Girl Lyrical Nanoha ViVid had Einhart to Vivio, and she was the Unknown Rival to Ixpellia in the beginning of the season, but Einhart has become a Friendly Enemy and the best friend of Vivio and she has given up on Ixpellia, who lies in a coma anyway. Many of the Inter-Middle Championship participants have a strong rivalry to each other. Harry vs. Victoria, Els vs. Harry, and many participants are rivaling Sieglinde, especially Micaiah has the strongest resolve to defeat her. Vivio gained another rival of the Friendly Enemy type, Miura Rinaldi, who has a similar role as Vita and Signum to Nanoha and Fate, respectively.
Magical Record Lyrical Nanoha Force has the Hückebein Family, who are opposing Special Duty Section 6. Among the Hückebein members who are rivals to several SDS6 members, there are Curren vs. Hayate, Veyron vs. Thoma, Cypha vs. Signum, and Arnage vs. Isis. It also looks like there will other up-coming rivalries between DeVille and Fate, and Rosalia and Lily.
Mazinger Z: Boss was The Rival from The Hero Kouji Kabuto. He was jealousy of Kouji because he was a better bike rider AND Sayaka had a crush on Kouji. The first thing he did when they met was picking a fight with him. However, as the plot and Character Development marched on, he slowly grew to respect him, and they eventually became friends (Boss still tried to compete with Kouji for Sayaka's heart, though). Also, in many Super Robot Wars games Kouji is rivals with Asuka Langley Soryu. The two of them always but heads. It is amusing, since she would be more liable to antagonize Tetsuya (both of them are two prideful Ace Pilots -Broken Aces, actually- can not stand losing).
Great Mazinger: Boss became again The Rival of The Hero -in this instance, Tetsuya Tsurugi- in the sequel. Boss could NOT stand Tetsuya, neither his arrogance (pretended arrogance, in fact; but Boss did not know that) nor Jun was infatuated with him. On the other hand, Tetsuya found Boss annoying, and he certainly did not care for Boss stalking Jun and showering her with unwanted attention. Tetsuya's true rival was Kouji, and when Kouji returned to the series, they got a very heated relationship. Kouji wanted to be friends or at least partners, but Tetsuya got very jealous and paranoid. He was an orphan kid with a huge Inferiority Superiority Complex, plenty abandonment issues and no self-esteem whatsoever, and he regarded Kouji like a threat would steal his father from him, take away his reason to exist, and replace him. Geez, he even accused Jun from wanting replacing him with Kouji (in spite of Jun barely knew Kouji and she had spent the whole series trying Tetsuya noticed her). He was ALWAYS arguing, bickering and fighting with Kouji, wether they were discussing tactics or battling the enemy.
UFO Robo Grendizer: Minister Zuril saw The Hero Duke fled like his rival, since he was in love with Rubina, King Vega's daughter, and Rubina was in love with Duke. Nonetheless, Duke regarded him like another Vegan commander he had to defeat.
Sasuke of Naruto floats in and out of this role. Kishimoto actually stated he read up on the archetype in various manga when making him.
Sasuke's entire creation was based on the fact that Kishimoto's editor said that Naruto needed a rival. Guess the fangirls should be grateful for this trope.
Then there's Sakura and Ino, though their rivalry has toned down a lot in Shippuden.
During the Chunin Exams, Rock Lee added three other ninjas as his other rivals besides his teammate Neji: Sasuke, Gaara and Naruto.
Also Guy and Kakashi, although this is arguably one-sided and usually Played for Laughs.
Obito and Kakashi, who had a very similar relationship like Naruto and Sasuke.
Jiraiya and Orochimaru, who had a very similar relationship like Naruto and Sasuke, too.
A hilarious case where One Piece creator Eiichiro Oda sees the Pretty Cure series as a rival due to his young children liking it more than One Piece. He vowed to take it down.
One could argue this for Luffy and Smoker, though it isn't so much as competing over anything as it is Luffy trying not to get caught by Smoker and get thrown into Impel Down. There's also Zoro and Mihawk, to the point that Mihawk trained him specifically for the job.
Gary (Motherfucking) Oak is Ash's Rival in Pokémon. In the anime he kept the rivalry until the end of the Johto saga, when he was defeated and pursued a different career, but he has since returned on a couple of occasions to help Ash and company, and is more of a Friendly Enemy than anything else.
The Hoenn arc was the only arc Ash himself went without a direct rival, although during The Hoenn League he went up against Morrison, who was much like Ash in his demeanor, and the two had a good battle.
In the Diamond and Pearl series, he has Paul, who happens to be at least a half-dozen shades darker than Gary ever was, to worry about, and the new guy is not only efficient in subduing Ash, but also darwinistically cruel; any Pokemon he deems weak is dropped or handed off without a second thought.
Ash later met Barry, who is a borderline ditz. Nevertheless, their rivalry is fairly friendly and Barry became a fan favorite. It also helps that he Lampshades most of the tropes/traditions in the show.
Bianca, Stephan, Cameron and Virgil are also introduced in this saga as rivals for Ash, the former two gaining more story and focus as rivals as it becomes clearer that the rivalry between Ash and Trip just wasn't working. Stephan and his Sawk even get their own rivals during the Clubsplosion arc.
Pokémon Special has Red and Green/Blue, mirroring the video games as well as Ash and Gary. They aren't ever explicitly hostile to one another, but are disdainful of their respective training techniques: Green thinks Red is airheaded and impulsive, relying on luck to win, while Red thinks Green is overly harsh and uncaring towards Pokemon, seeing them as simple tools. In the end, they wind up learning from each other (Red becomes calmer and more strategic, Green becomes more openly friendly with his Pokemon), and the two are Fire-Forged Friends as of the Pokemon League battles.
Red: Someone once told me that you have to know your limits! *recalls his exhausted Pokemon*
Green: Heh. And someone once told me that there is no victory in defeating a weakened opponent. Choose your next Pokemon, Red!
In The Prince of Tennis both Sanada and Atobe start out as formidable, intimidating opponents. Along the way they earn the main team's respect to a degree, and in the last match, they help the main character regain his memories so he can face his last opponent (who happens to be Sanada's captain).
Ryoga is easily Ranma's foremost rival and closest competitor (as seen here) throughout the series. They've fought several times (excluding minor tussles) and the outcome, while almost always with Ranma being the victor, are usually close. Ranma openly acknowledged Ryoga as his rival, during the "Mark of the Battling God" arc, and has also acknowledged that few could force him to fight seriously, or push him as hard as Ryoga can.
One-time manga character Asuka Saginomiya, "The White Lily", is Kodachi Kuno's rival. She IS Kodachi, just in a different color. She actually manages to top Kodachi in terms of hamming it up by carrying around a large picture frame with lilies tied to her back so she can pose.
s-CRY-ed is largely built around the rivalry between brash, free-spirited Kazuma and stoic, duty-obsessed Ryuhou. In other series it's usually The Power of Love that can never be extinguished and can overcome any obstacle; in s-CRY-ed it's Power of Hot-Blooded Macho Rivalry instead. It's a notable example for just how much effort is put into developing their rivalry: at first it's simply because Kazyma and Ryuhou piss each other off, but gradually, even as their viewpoints and personalities change and develop their rivalry simply intensifies over time rather then ever really reconciling, and the series ends on their climactic final battle.
Amusingly, in one episode a character with the ability to write a script that everyone must then follow as long as they're in it attempts to have the two reconcile in a friendly sparring match, his story ending in them becoming friends after working through their rivalry. It completely backfires, as Kazuma hates Ryuhou so much that just "fighting until they turn into friends" ISN'T enough for him, and it breaks the script and reality re-asserts itself.
Soul Eater has Ox claiming himself Maka's rival (only academically speaking), and Black Star sees himself as Kid's. Gopher appears to see Maka as his rival for Noah's attention, much to the girl's bewilderment upon first meeting the guy out to kill her with extreme prejudice. He takes the same attitude towards Kid who, like with Black Star, is simply not bothered.
Ultra Magnus to Optimus Prime in Transformers: Robots In Disguise, as he feels he should have gotten the Matrix. He never turns evil, despite Megatron's best efforts.
Seto Kaiba from Yu-Gi-Oh! practically defines The Rival.
Yu-Gi-Oh! GX has Jun Manjoume, a partial Expy of Kaiba, who serves as Judai's rival. Later, he gains a second rival (also a partial Expy of Kaiba, though utilizing different aspects of his personality) in Edo Phoenix. And for a few episodes Hell Kaiser Ryo could also count as a rival to both Judai and Sho. Also a Kaiba Expy, even using what is in its core a Cyber version of Kaibas deck.
A far more dangerous example in 5D's: three of the Dark Signers opposed one of the Signers for personal reasons. Kiryu to Yusei, Carly to Jack, and Misty to Aki. However, this was always due to their misinterpreting of past events (and Blatant Lies told to them), and each case ended with a truce called.
This continues in Yu Gi Oh Zexal. Shark was Yuma's first rival, later becoming a Friendly Enemy and occasional ally. Kaito is another rival and might be considered an Expy of Kaiba, but had much darker motives and could not be considered anything more than a Well-Intentioned Extremist until the end of the first season. (Kaito usually considered Astral to be more of a rival to him than Yuma was.)
Konjiki no Gash Bell!! has a number of these, most notably Brago, who is called "the demon to beat" by Kiyomaro.
They also have Bari who decisively beat them but spared their book. There was also Shinichi/Eshros, who fought the pair into a corner and would have won. Fortunately, Shinichi just wanted to test his strength against Kiyo and Zatch, and retrned their book.