With Elemental Rock-Paper-Scissors, that means everyone and thing has strengths and vulnerabilities to everything else. Except for those that are "outside" the system, that is.
A Non Elemental spell, power or monster will be the elemental equivalent of the Jack of All Stats: no weaknesses, but no noteworthy strength. Despite this "averageness", the Non Elemental's greatest strength is that he can bypass most elemental resistances and strengths, making him effective against elements or types that have no or rare weaknesses. That he's so well balanced means he can work in just about any situation on top of that, too.
Thematically, a Non Elemental character may be described as using raw Ki or Mana, or some fundamental cosmic force at the center of the elemental system. Alternately, he may have developed a power that somehow bypasses the element system or fuses them all together. For these reasons, Non Elemental characters and powers are sometimes considered advanced magic that is hard to master or acquire, and may be valued as Game Breakers. They may cross over with Infinity+1 Element due to this. Note that Non Elemental powers are usually, but not always, distinct from "physical" attacks.
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Sieg's alignment in Rave Master. It's implied that this is why he was able to master all 8 elements at such a young age (whereas others who seem near his age can control maybe two or three) but never confirmed.
Artifacts and artifact creatures are generally colorless in Magic: The Gathering. They usually use no colored mana, and so can be used in nearly any deck. Not all artifacts are colorless, though; the Shards of Alara, Scars of Mirrodin, and Theros blocks (plus one card from Future Sight) all include colored artifacts. There are also other colorless spells like most of the Eldrazi.
Force effects in AD&D2 and D&D3. Nothing was resistant to force, and force can even hit ghosts and other ethereal creatures as if they were solid. To compensate, force spells usually did less damage for their level. As a whole, too, Force magic isn't as focused as Abjuration or Elemental Fire, but has a good mix of attack, protection and utility spells; AD&D2.5 acknowledged Force as a school of effect, allowing specialists.
Sonic in D&D3 might be considered the poor man's substitute for Force in this respect. While it was an 'element' and thus could be resisted, enemies with natural reistances were very rare, and unless you had specific foreknowledge, no-one would prepare for a fight by bothering with putting up defenses against sonic. Like force, sonic damage also was generally smaller than the more traditional elements.
Magic Missile is notable as well. Pure Magic, hits the target automatically. Therefore, scales into higher levels surprisingly well.
The best example in third edition, although widely disallowed for a variety of reasons, is the Crystal Shard psionic power. Although the fluff describes it as a sharp crystal shard that deals piercing damage, and should thus be a part of the physical combat system that suffers from Damage Reduction (except for DR that is overcome solely by piercing damage, of course), technically all psionics are magic and thus bypass all DR entirely (in addition to bypassing all Spell Resistance by virtue of being an actual physical object). It scales very well and remains viable for as long as Hit Points matter in the game.
The true example for 3.x: Nameless damage. No type of damage is stated and it is considered to be of no type. However, it has to be noted that many nameless attacks merely behave as an obscure damage type: for instance, Searing Light and Holy Smite are nameless, but have their own rules as to what creatures are vulnerable/resistant to them and their types are referred to indirectly via description ("light-based attacks" and "Good aligned attacks"). Later on, Light and Holy became official damage types.
D&D 3.x and 4e also have a system of types for bonuses (like competency bonuses, sacred bonuses etc.) for determining whether they stack (you cannot add two enhancement bonuses to the same thing, only take the higher one, but you can add enhancement and insight for instance). Nameless bonuses exist and stack with everything except getting the same bonus twice (such as casting the same spell multiple times) thus every effect that can give you a bonus essentially has its own unique type.
Final Fantasy. Non-Elemental is the default for physical attacks, which aren't going to have an element unless you're using a specific weapon or skill. Elemental is the default for early direct-damage magic however, though a good number of spells (such as Standard Status Effects) aren't going to be elemental, either. Notably, most of the strongest direct-damage spells, like Ultima, Bahamut, Flare (unless a game makes it a fire element spell), and Meteor, are non-elemental.
In later Final Fantasy games, physical attacks could be considered elemental as some monsters are immune to them.
At least one Final Fantasy game also included hidden (or at least never mentioned in-game) physical elements related to the kind of weapon used (slashing, bashing, piercing, etc) that occasionally an enemy's defenses would be strong or weak against. The actual effect was trivial enough that many players wouldn't even notice, but it was there in the game data.
Normal humans (non-adepts) in Golden Sun. Adepts are canonically weak to the element opposing the one they control (though it doesn't really show with the player characters, whose resistances to the elements they're not aligned to are mostly equal and fixable with Djinn) and monsters have weaknesses and resistances all over the place, but regular humans are equally affected by all elements. Golden Sun: Dark Dawn introduced "dark" monsters, which are also equally vulnerable to all four elements.
Damage and Drain effects in The Elder Scrolls games are non-elemental Destruction effects. Damage is more expensive than its elemental equivalents, while Drain only causes temporary harm for the duration of the effect.
In Eternal Darkness, spells empowered with the Mantarok rune fall outside of the Elemental Rock-Paper-Scissors game of the other three ancients. Additionally, Mantarok's spells typically have special effects like damage over time.
Shin Megami Tensei games have "Almighty" spells. At minumum you'll typically encounter (in order of power) Megido, Megidola, and Megidolaon. Nothing can block them (though you can certainly resist them), but nothing is weak against them either. Almighty Spells also tend to have inflated MP costs for their damage output, making them bad MP economy.
In Persona 2, Almighty attacks are non elemental magic, which is elemental since some demons and personas are weak or strong to all type of magics when it's not all type of attacks. Even a few opponents are specifically immune to Almighty attacks, like Dark Alice and the Big Bad first form. That being said, there are a few attacks with no element at all like Another Dimension or Death Roulette. Most of them are instakill attacks.
Played straight in Persona 3 and averted in Persona 4. Both games retain the Megido series of Almighty spells, and performing an all-out attack with your whole party is an Almighty attack too. Unfortunately, in the latter game, Almighty attacks become just another elemental type, and some common enemies are resistant to it, making it hard to sweep the board with a team attack.
On another note, Shinjiro Aragaki in Persona 3 has the honor of bearing the only Persona with no elemental strengths or weaknesses.
Not exactly. In Persona 4, the All-Out attacks are a mix of physical and almighty so an enemy nulling physical attacks can't null the All-Out attack but a monster resisting physical can resist an All-Out attack.
In the first Digital Devil Saga, three enemies have some kind of resistance against almighty attacks : Bat is immune to all non earth damage while he take a defensive stance although using an almighty attack won't make you loose extra press turns, the optional boss Huang Long is immune to all attacks while he uses his defensive move and the Big Bad's final form and its orbs resist almighty damage, encouraging you to take down the orbs and be able to hurt the Big Bad with elemental or physical attacks.
In the second Digital Devil Saga, an Old Save Bonus can allow you to recieve a ring that resist Almighty damage.
In the DS spinoff Devil Survivor they are joined by Holy Dance and Drain, with Holy Dance being one of the best boss killing spells in the game. A grand total of one enemy resists Almighty damage, namely Beldr, who is immune to everything but mistletoe.
Devil Survivor 2 has the extremely rare Anti-Almighty skill, and a certain demon (Lucifer) and one of your possible allies (Yamato) both have built-in Almighty immunities.
Etrian Odyssey, another RPG made by Atlus, has the Megido and Eschaton spells, which deal non-elemental damage. When increasing the power of Megido with skill points, the effect of more skill points is listed as "Almighty Damage Up", a nod to SMT.
Normal is notable for having the least amount of weaknesses/resistances combined and Normal attacks aren't super-effective against anything. This makes a Normal-type a good choice to start a battle with if you have no idea what you're up against. Dragon is a good runner-up.
Whitney, Norman, Lenora, and Cheren are all Gym Leaders who specialize in Normal-Type Pokemon.
A few non-standard attacks in the game, such as Struggle, Future Sight and Doom Desire, are considered by the game engine to be typeless (though the latter two aren't as of Gen V), such that they are never super-effective and even do damage that bypasses Shedinja's Wonder Guard. Secondary damage sources, like confusion, also ignore type.
Set damage attacks Night Shade and Seismic Toss deal damage equal to the user's level regardless of resistances (though they still do no damage to opponents that are immune). Also, Dragon Rage always deals 40 damage, and Sonicboom deals 20, although it doesn't hit Ghosts.
In Pokemon Colosseum, Shadow Rush (the sole "corrupt" Shadow move in the game) functioned as non-elemental as well. In the sequel, it became the Infinity+1 Element and was doubly effective against everything except other Shadow Pokemon.
But with the power of most Shadow moves, super-effective for them is roughly equivalent to most normal-effective non-Shadow moves. The net effect is to make Shadow Pokemon resistant to each other while doing fairly normal damage to everything else.
In the Pokémon Mystery Dungeon seriess, Pokemon have the ability to perform a weak basic attack, which isn't of any typing. There's also two TMs exclusive to the series, Wide Slash and Vacuum Cut, which are both non-elemental. Wide Slash is a physical move that hits three tiles directly in front of the user, and Vacuum Cut is a set-damage attack that hits everything in a room.
The Blast spell in the Wild ARMs series of games. At least in the ones where Ley Points aren't a part of the battle system. In the games where they are, Blast takes the element of the hex the caster is standing and is only non-elemental if the hex has no such elemental traits.
Chaos type damage in Warcraft III does full damage to every armor type, including the heavily-armored Fortified and the otherwise nigh invulnerable Divine armor type.
Conversely, the "medium" armor type, before a game update, received the same amount of damage from all weapons.
Geppetto in Shadow Hearts: Covenant is the one character without an innate element. His elemental affliation depends on the dress his puppet, Cornelia, wears. In From The New World, Johnny, the Supporting Protagonist, doesn't have his own element; Shania, the game's Fusion user, is Darkness (as was Yuri from the previous game, also a Fusion user).
In Kingdom Hearts: Chain of Memories, most attacks are "physical" or fire/ice/thunder. But certain spells and most keyblade sleights are considered to be "special", which mostly work like this. 'Mostly', as a couple of bosses resist or are weak to it, and it pierces defences that regular elements can't get past.
Kingdom Hearts 3D has the Balloon line of spells and any wind spells. The former is commonly mistaken to be Water element, as it gives Water resistance in Dream Eater creation.
Jeanne d'Arc allows every single character to take on one of the three alignments Sol, Luna and Stella by equipping accessories of the respective elements. However, one can also allow these characters to have a neutral alignment by not equipping them with these accessories. In fact, most boss characters in the game have neutral alignment.
Diablo 2 has non-elemental magic damage, which very few enemies have resistance to. All seven character classes except the Druid and Sorceress have at least one skill for dealing magic damage, and there are some unique items that add magic damage to your attacks.
In the Ultima series of games (at least later on), Lightning damage fills this role, because no monster has an innate resistance to it. This is how Glass Swords are able to always deal maximum damage; they deal 255 points of Lightning damage.
Tales Series games often feature one or two non-elemental summon spells, usually Maxwell or Origin, as well as a few non-elemental attack spells like Tractor Beam and Meteor Storm. In Tales of Eternia, though, Maxwell was actually "Elemental elemental", that is, embodying all elements.
Then again, every "non-elemental" weapon in the game is also "elemental-elemental", so it still counts.
The Disgaea series have three regular elements available to the mage classes: Fire, Ice, Wind. They also have a fourth non-elemental spell type: Star.
Though occasionally Star will count as an element that just has no weaknesses or resistances, for example, the White Dragon monster class in Disgaea 2 is immune to "non-elemental special abilities," but is affected by anything with the Star element. While most of the abilities the immunity ends up including are physical, there are several monster classes with non-element magic attacks, including the White Dragons themselves.
Also, an elemental spell against a monster weak to that element does more damage than a star elemental spell against the same monster: star does better than average damage, but it doesn't do the best possible damage.
The three types of weapons (swords/blades, axes/hammers, and lances/polearms) and the three elements of magic (fire, wind, and lightning) used in Fire Emblem games, all adhere to an Elemental Rock-Paper-Scissors mechanic. The exception is bows for the weapon triangle which have no inherent advantage or disadvantage over the other weapons. A few of the games will also place Light magic outside the Trinity of Magic, though just as frequently gives it a different spot in the Trinity (Anima -> Light -> Dark -> Anima).
Void doesn't always work, though. There's always things like the Void Dragon healed by it.
Another Non-Elemental would be the Death Element. Take a guess as to what it does.
The new King's Bounty games have a few abilities that deal astral damage. With the exception of Death Star, they are Rage-based and only hit enemies (Death Star hits all living beings instead). Holy may also count, considering that there is no dedicated form of protection against it, but it's not very effective against anything but the undead and demons.
In World of Warcraft, there are six elemental schools of damage (holy, shadow, frost, fire, nature, arcane) plus a seventh category called physical. Of these, in earlier iterations of World of Warcraft only Holy damage had no form of resistance stat (armor was resistance against physical damage, shadow resistance reduced shadow damage, fire resistance reduced fire, etc.)
In more recent updates of the game, spell class resistance stats on item have mostly been eliminated except in special cases, but there are various other forms of resistances through various game mechanics.
Furthermore, many spells are categorized by two or more spell types (shadowfrost, shadowflame, frostfire, etc.) and always do damage based on the lower of the target's resistances; there is one category called 'Magic Damage' which is classed as all six spell types, and another category called 'Chaos Damage' which is all six spell types plus physical damage, which always in all cases benefits the lowest target's lowest resistance.
A lot of boss abilities, and some abilities used by GMs for testing purposes, don't fall into categories at all so that they can't be dodged, parried, blocked, absorbed, deflected, reflected, resisted, interrupted or dispelled.
In Final Fantasy VII has a Good Bad Bug example with Ultima Weapon's "Ultima Beam" attack, which seems non-elemental but is actually of the "Hidden" element (it's the only offensive move with said element, the others are just status moves of various sorts). It's possible to resist or even absorb it by attaching Elemental to any non-elemental materia, though Ultima Weapon is easy enough that you probably don't need to bother.
Bahamut Lagoon features Uni magic, the element of black dragons. It's typeless, and does approximately twice as much damage as normal attacks, but always has a 50% chance of failure. It also makes Mini-Devil random attacks (which already have a random chance of failure or negative effect) become 'extremely' random.
Spiral Knights has the Normal damage type, which damages all enemy types equally.
Early on City of Heroes had a few sources of untyped damage, mostly this was later replaced by toxic damage, but as a legacy effect toxic has no associated typed defence stat. (you can still acquire resistance normally though)
In the MOTHER series (except the first game), the powerful non-elemental attack is the trademark of The Hero of each game, where it is named after their favorite thing. Ness's attack is called PSI/PK Rockin' while Lucas's is called PK Love.
PK Starstorm, learned by Poo and Kumatora, is also non-elemental and in terms of its damage to cost ratio, the best PSI/PK attack in the game.
In the iOS (iPod, iPad, iPhone) game Geomon, the recently introduced Mech type has no alignment, but no pure-mech Mon has been introduced.
While lacking a Elemental Rock-Paper-Scissors system, Path of Exile's Chaos element pulls double duty as both this and Armor Piercing. It has none of the Standard Status Effects that fire, ice and lightning possess, but it goes straight through energy shields and damages health directly. It's also the most difficult element to defend against as the only way to get chaos resistance is through nodes in the game's massive passive skill grid. There's even a passive skill that grants immunity to said element, but turns the player into a One-Hit-Point Wonder in return.
Card-based RPG Lost Kingdoms has the four typical elements, along with neutral type. Cards and monsters that dealt with pure magic, death, or darkness were categorized as neutral. In Lost Kingdoms 2, the player could even specialize in neutral, but neutral affinity increased much slower than other elements (besides the new steel element).
The Epic Battle Fantasy series has a few abilities and weapons with this property. They aren't usually very good except against enemies that resist most things.
An example appears in the Grand Finale of Avatar: The Last Airbender : Spirit/Energy Bending, which Aang uses to completely depower the Big Bad. Despite looking like a heaping helping of Deus ex Machina at first, it makes sense when you remember that the Avatar is both a spirit and a vital part of the Universe, so souls and cosmic energy are well within his power to command and manipulate. Sure, it's done rarely and uses a ton of power, but it can be done. Lion Turtles, ancient beings that predate the Avatar Cycle itself, were masters of Energybending. One of them taught Aang how to do it.