All the shamans in Shaman King have this to some degree. Hao, the most enlightened and powerful of all, decided that to create utopia all the ignorant humans have to die, leaving only shamans to populate the world.
After years of contemplation, fallen Buddhist monk Anji of Rurouni Kenshin learned the principle that allowed him to develop the Futae no Kiwami attack, a powerful punch that can turn rocks into dust.
Senjutsu from Naruto works like this by requiring the user to be In Harmony with Nature and combine the physical and spiritual energy found in the body with natural energy found in outside it in nature.
Zanpakutou-based abilities are this in Bleach. Zanpakutou represent the user's specific talents and strengths, and you must form a bond with them to access all of your hidden powers. The problem is that they also represent your faults and weaknesses, too, and the parts of your personality you don't want to acknowledge.
This was the entire premise of the Charlton Comics character Peter Cannon, Thunderbolt. His Watchmen equivalent, Ozymandias, doesn't quite fit the archetype, though.
Most comic book magicians qualify, at least the ones who don't go down the "worship dark gods/sell your soul" paths. Doctor Strange, for example, or for a much earlier example, the Green Lama.
Zora in the comic Powers is an inversion. By rejecting all spirituality and realizing she is "her own god", she gained godlike powers.
A good half of Kalimán's powers came from his intense mental and spiritual training in the orient. The other half? Sheer Badassitude.
In Spawn, Harry Houdini discovered real magic in a moment of extreme focus required to pull off one of his escapologist tricks.
The Contemplator, one of the Elders of the Universe in the Marvel Universe. His study of meditation, philosophy and mental/spiritual development gave him powers such as telepathy, mental domination of others, precognition, telekinesis, levitation, and astral projection.
David X in Casanova went into meditative seclusion as a performance artist. Twelve years later, he emerged as a low-grade Physical God.
Superhero Johnny Quick and his daughter Jesse both gain their Super Speed from reciting a formula describing a 4th dimensional object while attempting to imagine said object. Later retconned into being their ways of accessing the Speed Force.
Neo. The many Buddhist and Christological allusions for Neo throughout the series range from subtlely added to pounding in your ears with its Sanskrit-based musical score in the final movie. When he becomes The One, also becomes a Reality Warper because he understands the underlayer of the Matrix. Outside of the Matrix, Neo's powers to force enemy Machines to explode seems more technological, attributed to how he is differently wired than other humans in regards to the Matrix and the Machines. He effectively cracked into Sentinels wirelessly, thus interrupting their actions. The ability is limited in scope and range, worked only on Sentinels and their bombs, and tired Neo greatly in the final battle on his way to the Machine City.
All of the disconnected humans who reentered as Zion operatives use this to bend and break the physical laws within the Matrix, but it had its limits.
The Oracle and her weird count because "There is no spoon"—that is, they mastered simple forms of Reality Warping.
Then there's Smith, the computer virus version of this, managing to infect disconnected humans. Luckily, he was defeated before finding out whether he could spread in the real world the way he did inside the Matrix.
Lamont Cranston in The Shadow learned how to "cloud men's minds", among other abilities. The end of the film implies that this training is used to unlock a normally-unused part of the brain. When Shiwan Khan gets that part of the brain removed by a surgeon who works for the Shadow, he can't Mind Control people anymore.
The Sphinx in the film version of Mystery Men is a parody of this character type; his superpower is "being terribly mysterious" and he speaks entirely in Ice-Cream Koan. He's also a straight example, since he really can slice guns in half with his mind as he claims.
In The Men Who Stare at Goats, the training doctrine of the New Earth Army was designed to trigger this. The soldiers were supposed to gain the powers of remote viewing, invisibility, walking through walls and the sparkling eyes technique while delving into Eastern mysticisim.
The mental/physical Schools of Frank Herbert's Dune exemplify this, especially the Bene Gesserit, who have attained complete control over their own bodies as well as other abilities that cause people to consider them to be "witches". It's downplayed because some of it is done by ingesting spice.
The members of these schools didn't see themselves as superpowered, simply enlightened enough to benefit from the training. The Kwizatz Haderach is the one whom they considered superpowered. And every generation or so there would be another enlightened further beyond.
In The Wheel of Time, Rand was always a scarily powerful Channeler, but it was plain that the rigors of his role and magic were taking a terrible toll on him, nearly driving him to insanity and nihilism until he has an epiphany at the end of Book 12 which brings him close to what might be called enlightenment, making him a messiah in truth as well as title and dramatically amping up his powers (though he's still weaker than the Big Bad, who's an actual god). His Evil Counterpart Moridin has what could be classed as an evil version of this- a true nihilist, he's the closest of all mortals to the Big Bad's goals and methods, and as such is the only one allowed to access his special magic system, the True Power. The True Power, though not any stronger than the One Power (which fuels most magic-users), is exempt from some of its restrictions and can do things it can't.
Variation in I, Jedi. Trainee Jedi Corran Horn has been unable to use telekinesis, a power which comes easily to most Jedi. He thinks he might be able to break through a mental barrier if he tries to do it with a huge rock, closing his eyes and really focusing. He pictures the rock rising up into the air, opens his eyes...to find the rock has stayed where it was, but everyone else is staring up in the air. He learns that his family have always been poor at telekinesis, but gifted at making illusions.
The humor book How To Be A Superhero describes "Becoming One With the Cosmos" as one source of super-powers, though the reader is warned not to Become One With New York City and turning into a massive pile of shit as a result.
Stargate SG-1 takes this approach: the closer one gets to Ascension (sort of a parallel for enlightenment, and somewhat based on Zen Buddhist belief), the more power over nature one gains. People on the cusp of Ascending were shown to gain psychic powers, heal people, etc., and fully Ascended beings had complete control over the natural universe. However, the downside is that no one can remain on the cusp of Ascending for a long time. It then becomes a Die or Fly case, in which the person must Ascend before their body dies. McKay in Stargate Atlantis manages to come up with an alternative solution just before his body expires.
A one-time foe on Angel: a girl who combined this with removing her own eyes, to gain the ability to see a second or so into the future. This is a useful ability in an assassin.
In Mage: The Ascension and Mage: The Awakening, the eponymous Mages get their reality warping magic thanks to an out of the blue epiphany regarding the nature of reality. This epiphany is natural, but has little discernible rhyme or reason or blood connection. The less powerful Thaumaturges potentially (it's complicated) derive the ability to use magic not from innate supernatural connections, but from intense and prolonged spiritual and occult study. Of course, this being the World Of Darkness spiritual enlightenment doesn't necessarily require morality.
In Vampire: The Masquerade, even the damned can find Enlightenment Superpowers. Saulot, a founder of one of the (now defunct) thirteen vampire clans, traveled to the Orient and studied under enlightened supernaturals, returning with mystic healing powers of mind and body... as well as a Third Eye all his children would inherit.
The successor, Vampire: The Requiem, has its own version. The Ordo Dracul is a covenant devoted to transcending the vampiric condition, just like their founder did. By studying and meditating on their condition, they gain access to Coils of the Dragon, powers that lessen their reliance on blood, allow them to pass as human, or give them a greater resistance to the unpleasant stimuli that often harrow other vampires.
Saulot studied under the Kuei-jin, the so-called Kindred of the East, who pursue various paths to enlightenment, seeking transcendence of their accursed state. In the process, they gain access to ever-greater powers.
In the obscure game "In Dark Alleys", which can best be described as "The Matrix meets J-Horror meets Kult", people with a powerful enough force of will and a clear enough understanding of reality (long story short, it's all in your mind) can break the laws of physics at a whim.
One type of bonus that can be gained in Third edition Dungeons & Dragons is an "insight bonus". Highly advanced beings who have such a bonus naturally (at least in their armor class), and not just because of some spell, are presumably examples of this trope; they include at least Demon Lords and Archdevils and the members of a Council of Angels, with the latter having higher such bonuses.
Also, D&D Monks. Spend enough time meditating and punching things to death, and you too can teleport or ignore disease and aging!
In RuneQuest/Hero Quest's Glorantha setting, Mystics can gain supernatural powers if they stop searching for enlightenment and get distracted. Many mystics are only being dedicated long enough to get the really good distractions. (See Buddhism under "Real Life", below.)
Exalted: The basic idea of charms is that its user know well enough on how to manipulate the Essence of the world. Exaltation speeds up this progress and helps you not dying in the training, but anyone can theoretically achieve Enlightened Essence if they have the conviction.
Supernatural Martial Arts on the other hand, means that you're enlightened to the Essence of your own body. The majority of sifu have enlightenment theme about them, but the Sidereals take the proverbial cake.
Then there is Sorcery, which means that you're enlightened enough to know the innermost workings of Essence motes so that you can break the law of physics. To put it in a comparison, mastery of charms puts you on the level of mechanics, but mastery of Sorcery puts you on the level of theoretical physicist.
Exalts raise their permanent Essence score through meditation and discipline, giving them access to more power and beefier Charms.
In Chuubo's Marvelous Wish-Granting Engine, Jasmine Apocynum saw the truth beyond the lie of the world, the True Thing underneath the tales of blood and bone and wood and dirt, after being hit in the head by a dodgeball. She saw that for just a moment, and then it went away. She became or expressed becoming a Rider of the Bleak Academy, which is sometimes a metaphor for bodhisattvas when it's not a metaphor for death. Afterward, she gained the ability to pull people's hearts out and reshape the flesh into a seething void of night, or a giant mecha. This is eminently logical, sensible, and sane.
Legends Of The Wulin has Enlightened Chi, which can only be gained by committing virtuous deeds in accordance with religious or philosophical moral codes. Everyone worth mentioning in the setting has Chi, though, this just makes some of your Chi more useful than normal.
Mages in Shadowrun can gain a greater understanding of the nature of magic through initiation, which allows them to learn to use magic in ways they couldn't before (e.g. altering their astral signatures, making them more resistant to hostile spells, or invoking more powerful spirits). Initiation also raises the cap on the Magic attribute, which not only allows them to become more powerful, but also offsets the reduction of the attribute from Essence loss. Technomancers have a similar technique called submersion, giving them a greater understanding of the Deep Resonance.
Great Tiger of the Punch-Out!! series, especially in the Wii version.
The Monk class in the MMORPG Dungeons & Dragons Online use ki to charge their attacks, leap great distances, heal friends or curse enemies. One variant can even kill in a single ki-juicy punch. The religious aspects of DDO gameplay, compared to its tabletop counterpart, are very watered down for Monks and other classes, however.
In the Fate/stay night visual novels, the teacher Kuzuki is one of the evil types - his perfect balance within himself and the world (even Saber expresses her surprise at his "perfect breathing") combine with an extremely esoteric martial arts style to enable him to hold his own even with a Servant in hand to hand combat, though he is not particularly concerned with the traditional trappings of enlightenment.
The Graybeards in Skyrim devote their lives to studying the Thu'um. This regimen is the only way mortals can learn and master the Thu'um. The Dragonborn doesn't have to go through this because he/she isn't an ordinary mortal.
The Pokémon Darmanitan is capable of changing its form if it has the ability Zen Mode. When its health drops below 50 percent, its stats change from a Glass Cannon to a Mighty Glacier (as well as increases in base stat total) and its type changes to Fire/Psychic and its physical attack and special attack are swapped.
Emperor Shaohao, introduced in the backstory for World of Warcraft Mists of Pandaria, purged himself of anger, hatred, doubt, despair, fear, and violence and sealed their physical manifestations away beneath the earth. Using his new-found power, Shaohao cloaked his land in a mist shield that protected it from the Great Sundering. It's downplayed because Shaohao was not able to expel his pride and it was his pride which powered the mists.
As of Soul Calibur V, Kilik has reached enlightenment, which accompanies a switch from a staff fighting style to mimicking every male fighter's style at random. The details of how this occurred are only available in the official Soul series artbook, but the line he delivers when performing any Critical Edge move makes it clear.
The story behind Supernatural Martial Arts of El Goonish Shive. Not unlike the great masters of old times, sensei Greg was a martial artist who concentrated his mind on an issue not related to anything in this world at all, for hours and hours, without a pause, until everything became clear. Yes, he invented the new style and one more approach to his world's Functional Magic. That being sensei Greg, this means he watched anime 168 hours non-stop and, accordingly, created "Anime-Style Martial Arts".
Rebecca Stone of the Whateley Universe. A kindly woman who learned the way of the Tao in the 1800's and is now acting as a mentor for Bladedancer. Rebecca was a nurse and is more focused on healing arts and magic than on being able to fight supervillains.
In Avatar: The Last Airbender, a guru reveals to Aang that, through spiritual enlightenment and releasing his chakras, he can control the Avatar State and go in and out at will. Before this point, Aang can only go into it if he's scared, angry, or psychically aligned with a vengeful water god.
A major theme in The Legend of Korra is the inability of Korra, the new Avatar and Aang's reincarnation, to access the spiritual powers associated with being the Avatar. She focuses on the physical side of bending, much to the dismay of her mentors.
Korra: I haven't ignored [the spiritual side of bending], it just... doesn't come as easy to me.
A villainous example: in the season finale of Book 3, Zaheer achieves the power of flight after the death of his lover P'li severs the last of his earthly bonds.
In My Little Pony: Friendship Is Magic, Twilight Sparkle's lessons about friendship eventually build up to her obtaining her alicorn form, which she obtained by realizing how much her friends meant to her and completing Star Swirl the Bearded's unfinished masterpiece spell.
The monks of the Chánnote Better known as the Japanese term, "Zen", in the west Buddhist temple of Shaolin are most well known for their martial arts skills, although this is somewhat of a case of real life Memetic Badass. Even so, Shaolin monks are extremely skilled martial artists and can do feats of strength, dexterity, and focus that seem superhuman — all due to spending tens of thousands of hours training in martial arts as a form of meditation.
While we're on the subject, meditation in general — Zazen, Yoga, etc. Zen Buddhism, for example, takes the position that if you can focus on something as boring as sitting in a room staring at nothing all day (Zazen), then you can focus on, say, the best path to get out of a moving car in a big hurry, or how to dodge a sucker punch at a bar. While it won't make a person superhuman, one should never underestimate the ability to focus when needed.
Some sects of Buddhism believe that a person will gain supernatural powers as he approaches enlightenment but if you go into meditation for the sole purpose of gaining these powers, it won't work. The powers are sort of a "side effect" of enlightenment; anyone who is that close to enlightenment knows he has no need of such things. There are statements from Gautama Buddha himself indicating that supernatural powers distract from enlightenment if obtained first, which prevents beings like gods and demons from becoming enlightened.
Theurgy and Theosophy in general, but you start the rituals before you've reached enlightenment.
Allegedly the monks of Tibet had these powers, but they had mysteriously disappeared by the time the British invaded in 1904 and the Chinese Communists in 1950.
Depending on what you call Enlightenment, all of science and technology. The Enlightenment in this sense may not be about meditation or subjective reality, but it's definitely the product of altering one's habits of thought, overcoming preconceived ideas, and learning the true nature of reality.