Aun Freya, who has the power to stop time in a wide area around her, actually lacks the secondary powers to prevent it from affecting herself as well (although she still has the required secondary powers to make sure the part of the world she's affecting still moves with the rest of the world, etc.). It's up to Photon, who is immune to just about anything to go in, knock her out to cancel it, and usually to beat the everloving crap out of whatever scared her so much she used her power.
Pointed out by the author in the manga version of Ghost in the Shell. He goes on to explain that this is why full-body prostheses, like those of Motoko Kusanagi (whose only remaining organic components are her brain and nervous system), are preferable. The manga points out that if someone were to get, for example, his arm replaced with a prosthetic, his biological body would still be subject to the strain of muscle fibers tearing where the arm is connected if they lift too much weight. Full-body prosthetics on the other hand have theoretically limitless strength (since it's now all a matter of technology), enabling them to jump huge distances (like how the Incredible Hulk does) and survive the impact of the landing (albiet the ground will take damage.)
Alphonse, as a soul inhabiting a suit of metal armor, has the advantages of Nigh-Invulnerability, but also the drawbacks. His great weight serves to anchor him, but also hinders him. He is unable to eat or sleep... but also has no need to. He cannot feel pain... because he is completely numb. However, while an inability to feel without nerve endings makes sense, for some reason he can still see without eyeballs, hear without eardrums, speak without vocal cords, and even move without muscles or feeling and equilibrium required to coordinate movement.
The speaking part is actually lampshaded a bit, in that Al's voice has not changed since the incident. In The Movie for the first anime, Conqueror of Shamballa, Al has gotten his body back, and grown up to the age he was in the series, and as such his voice is deeper. Oddly, when he gets his soul temporarily shifted to an identical suit of armor on Earth, he sounds like a nine-year-old again.
The manga does explain why he can still remember things without a brain his soul is still "connected" to his body and mind which is on the other side of the Gate (and still aging).
Al and the other animated armors can move without muscles because his soul is infused into the armor. During Ed's fight with Number 48, 48 shows Ed the blood seal on the inside of his helmet. After some fighting Ed knocks 48's head off and thinks he's won because the part of the armor the seal is on is no longer connected to the main body. However, it's revealed that he has another seal on the chest of his armor, but after Ed uses a variant of Scar's disintegration to destroy part of 48's armor, the severed torso with the seal is no longer able to move its legs or defend itself against Ed.
Similarly, when Barry the Chopper's armor is destroyed, the piece with the blood seal on it can still talk, hear, see, and move (he can't move a lot, but he can move), but the rest of the armor can't.
Still, it is shown that somebody in Al's situation will feel it if you touch the blood seal. At one point in the anime, Lust scratches Al's blood seal and he seemed to be in pain.
Alchemy outwardly seems to change objects from one state to another, but the process is basically breaking things down and rebuilding them, not outright changing them as-is. This is why Scar can only destroy things with his "alchemy": He does not know how to rebuild things once he "breaks them down". Until the end of the series, where months of studying his brother's notes allows him to apply the "rebuild" tattoo to his other arm - which comes in very handy during his climactic fight against King Bradley.
Brook from One Piece is basically made out of these. In life, he ate a fruit that allowed him to return to life once after he died, aside from this he only had the powers of a human. When he died, the ship he was on was in a dense fog and it took him a year for his soul to locate his body and revive. Because of the time, his body decomposed until all that was left was a skeleton with an afro (strong roots). Despite reviving being the main power of the fruit, it also provided him with everything he needed to survive in his body; he can move the bones without muscles, see without eyes, speak (and drown) without lungs, eat without a stomach, use a toilet without bowels (Luffy asked), etc. Many of Brook's ACTUAL powers are merely extensions of this. For example, in life he was over 8 feet tall and pretty strong (if not super strong). He maintains that same level of strength and can even increase it with time, but now his body is a mere fraction of the weight, allowing him to leap incredible distances and run across water.
Luffy can bleed so much without dying because one of his secondary powers is a circulatory system that can make extra blood instantaneously to pump into his stretched out limbs. In fact, his rubberized organs, bones, and indeed every single vital part of his anatomy is the only reason he is able to withstand the stresses inherent in his Gear Second Super Mode. According to Rob Lucci, if anyone but Luffy were to use something like Gear Second, their heart would explode.
Haki users can develop armament and sensory haki (or King's haki if they have the disposition needed) but these simply make their skin harder, give them super reflexes or the ability to knock people out with their aura. In Luffy's fight with the Snake sisters Marigold and Sonia who could use armament and sensory respectively neither of them could beat Luffy's gear second for these reasons: While Luffy's blow didn't hurt Marigold as much it still knocked her back because she couldn't anchor herself and Sonia couldn't keep up with Luffy's moves even though she knew where he was going to be.
Logia users seem to receive a lot of secondary powers too such as Ace, Akainu and Aokiji being temperature resistant and Kizaru having reflexes fast enough to keep up with his light speed movement.
In Bleach, this trope is partially acknowledged in that Ichigo's speed increase from his Bankai lacks one notable secondary power: namely the ability to withstand his own speed. His first use of Bankai almost causes his own bones to break towards the end of the fight. He has to toughen up before he can get over that particular limitation.
Word of God also states that the reason Ichigo and Renji can throw around their huge swords with no effort is because the swords don't weigh anything to them as they are a part of their soul, but they sure as hell weigh a lot to the person on the receiving end
There's also the scene where Kurotsuchi Mayuri inflicts a drug on his opponent that increases his perception of time, resulting in the world appearing to slow around him, without actually granting him super reflexes so his body can keep up (Mayuri mentions that he usually dilutes this particular drug). As a result, poor Szayel can do nothing to stop Mayuri from stabbing him in the heart, despite seeing it coming for what to him seems like centuries. By the end, Szayel is begging for death.
Required Secondary Powers also proved key to defeating an Espada, Barragan, whose power was that his touch could rot and decay anything. His opponent, Hachigen, worked out that he had to have a secondary power to keep the rotting touch away from his own body. Hachi exploited that fact by warping his own decaying arm inside the Espada's body, causing Barragan to rot away from the inside out.
When training to use his Full bring Ichigo needs to undergo intense endurance and strength training as his time spent as a Shinigami did nothing for his physical strength which his Full bring needs.
Captains and Lieutenants also are given power limiters when in the human world and compress their swords to more manageable sizes. When Aizen (after achieving near-Physical God level of power) approaches a human without one and Grand Fisher fights Issin Kurosaki we find out why.
In Dragon Ball Z, the weakest member of the Ginyu Force has the power to stop time. However, he can't maintain it indefinitely and can only do so for as long as he can hold his breath and maintains the proper concentration. Considering how out of shape he is, this isn't long.
Fridge Logic dictates this is why Oozaru form increases a Saiyan's strength and ki ten-fold. Growing to giant proportions allows them to access additional strength and ki, which would be needed to prevent their body collapsing under its own weight.
Many characters of Dragon Ball and Dragon Ball Z have super anchoring abilities as addition to their strength. Especially noticeable with Tao Pai Pai, in his Curb-Stomp Battle with Bora. He holds onto Bora's spear, who is unable to even move it one bit, not even upwards even though he should by all means be able to lift Tao Pai Pai upwards as well. Fridge Brilliance comes into play here, these characters fly by pushing ki in the proper directions. So presumably they could do something similar to hold themselves in place.
For Goku and other, Super forms (Kaioken, Super Saiyajin, etc.) tends to put MASSIVE strain on the users body, leaving them tired and in pain as soon as the transformation ends.
The manga series Cannon God Exaxxion goes into some considerable detail about what a 500+ ft. Humongous Mecha would need to be able to move around. The mechs use Imported Alien Phlebotinum gravity/inertial control devices to prevent them from collapsing under their own weight & allow their limbs to move at a reasonable speed, instead of feet they have invisible forcefields to distribute their weight to prevent them from crashing through the ground, because having actual feet big enough to accomplish this herculean task would look insanely silly (the fields are described as "invisible snowshoes" at one point), although at least that way bystanders would see them coming rather than being crushed underfoot by a robot that appears to be safely dozens of feet away. Aside from this, they're otherwise portrayed realistically & thus cause massive property damage to anything they get anywhere near simply by walking around.
Another contractor had bursts of super-speed, complete with friction resistance and clothes-extension. Unfortunately the friction-resistance only worked on air: Cue nastiness from Newton's Third Law as he ran into some raindrops at a speed faster than a speeding bullet.
There's also an episode where Hei faces down another Contractor with powers like his. When they grab each other, they form a complete circuit, knocking each other back but not producing a lethal shock.
This is far from the only fun the show has with his powers- most of the issues listed above are addressed simply by requiring something to conduct electricity. Plus, that's the only way he can control the path at all; shocking someone through a puddle of water on the road also blew out a nearby streetlight, for example. The finale addresses the manipulate electrons = manipulate matter issue, too. Turns out that in the right place, BK-201 is an outright Reality Warper.
Nancy Makuhari (a.k.a. "Miss Deep") of Read or Die possesses the power of intangibility, but is unable to breathe while using her powers, as air passes through her lungs while she is intangible. This presents the very real danger of drowning while using her powers underwater.
This is referenced in Psycho Busters when lead Kakeru wonders whether Kaito ever gets burned by his own flames. He then answers himself saying "Nah! He must have the impervious-to-fire power!". Kakeru later sees Kaito turning down a coffee saying his tongue is sensitive to heat.
Accelerator in A Certain Magical Index has a rather interesting required secondary power: mathematics and spatial relations. Yes, without them he could probably deflect anything anyone sent at him but he couldn't really aim. Vector control requires him to be able to figure out what to actually do to get the result he wants. Post Heel-Face Turn he's notably weaker because of some brain damage.
In fact, it's mentioned that most powers, if not all, are based on this, revealing the reason why the ability users have to study extensively to be able to get more powerful.
Some of the others are brought up in the A Certain Scientific Railgun side story. Range and accuracy are the limiting factors of Kuroko's teleporter abilities (any objects intersecting with things she have teleported are explicitly stated to be displaced). Mikoto uses the heat generated from her electrical abilities on two separate occasions, once to cook an opponent using a non-conductive barrier and another time on a smaller scale to cook rice in a metal canteen when the stove isn't working. In the anime Uiharu mentions a specific lack of required extras. She has the power to prevent things from changing temperature but her hands can only tolerate a normal range of temperatures limiting the potential for its use (though she's great at keeping take-away warm). Touma also has some going for him otherwise attacks like Mikoto's railgun would still be lethal as momentum keeps the projectile going even after her power has been negated.
Shizuri Mugino, the Meltdowner, can fire laser beams and create energy constructs like shields, but is not immune to them, leading to Shiage Hamazura tricking her into blowing herself up.
Maria Kumokawa can increase the centrifugal force her movements generate, greatly enhancing her Dance Battler style. Since she does not have enhanced durability, she rarely goes over double the force, to minimize the chances of her breaking a bone or otherwise injuring herself.
According the the backstory of Zeta Gundam, Transforming Mecha are only possible, or at least practical, due to a revolutionary variable-friction magnetic coating on their joints that was first used by the original RX-78 series Gundams. This allows them to change in a snap. Without it, their joints would either be too rigid & their Transformation Sequence would take hours, or else their limbs & other moving parts would constantly be flopping around & falling out from under them.
Similar to the above, the various signature Transforming Mecha of Superdimension Fortress Macross and its sequels employ 'Energy Conversion Armor' which uses metals and technologies reverse engineered from the eponymous Macross itself (termed 'Overtechnology'). This armor can absorb energy and use that energy to reinforce it's own structural integrity. According to the various supplementary materials of the series, this is what the energy output of the Reaction Engines is used for when not being 100% committed to thrust. Thus when in Battroid (giant robot) mode, standing on the ground, with 0 thrust, 100% of the engine output is going towards telling the Square/Cube Law to take a hike.
In Naruto the Fourth Hokage's "Flying Thunder God" technique lets him teleport to within a certain distance of a seal which he puts on weapons and enemies, thus avoiding Tele Fragging by only teleporting to places likely to be close to open space. Likewise Tobi once implied that he can detect things (or at least people) from great distances which would probably explain why he never teleports into random objects.
In a case of Fridge Brilliance, the ability to warp to several targets at once within seconds with the jutsu would require a good dose of Hyper Awareness. Otherwise, the user's perception of time and space would be screwed up badly. Worst case scenario, if the user fails to do this, their bodies could be wrecked.
It's stated that the Super Speed charge the Chidori is used at would make it impossible/impractical because it's so fast that it causes a tunnel vision effect if Kakashi and Sasuke didn't have their Sharigan eyes to speed up their reflexes. The same goes for Kiba and Akamaru's Dual Wolf Fang (which is even harder for them to see while using since it also has them spinning the whole time), so they need to have Akamaru piss on the enemy beforehand, so they can find their target by scent.
The sharingan itself has a similar problem, it grants the user superhuman perception but not superhuman agility. Sometimes a sharingan user can predict an attack and see exactly where it's coming from, but if their body can't keep up with their opponent they still won't be able to dodge/block in time.
Another example is Naruto's Rasenshuriken jutsu. At first his own arm would get busted up due to recoil damage, since it's a Rasengan-based technique and sits on the palm of his hand. After he gained Sage mode, the enhanced chakra gave Naruto the extra powers he needed to be able to throw the damn thing.
Also, most of his Rasengan techniques require this. Due to his poor chakra control, he relies on his clones to assist in properly forming even a basic Rasengan. He eventually removes the need when obtains a different secondary super power, the ability to form chakra arms with the Kyuubi's chakra.
Naruto's Healing Factor also qualifies. It's based on rapid mitosis, which shortens his life every time he uses it. However, he has longevity on his side- members of his clan are very long lived, so he can still live to a ripe old age even after having his skin burned off multiple times.
In fact, every Nine Tails Jinchuriki has been from his clan, precisely because they have the secondary powers to handle it.
Sakon and Ukon's ability to fuse their bodies with each other or other people requires them to be able to greatly manipulate the structure of their own body in some other ways, including healing themselves.
Susanoo generates a very strong Force Field, but it lacks anchoring (its user has been both knocked into another room, although not injured, and nearly sucked into the mouth of a giant monster) and at least slightly air permeable (an acidic mist was able to get through it, albeit it's still better than being unprotected).
The databooks state that the Earth technique to burrow underground has the user act as a human compass to keep track of their location and sense what's above so they know where to attack when coming up.
Mabui's ability to move a target from one location to another at the speed of light is largely limited to inanimate objects. For a living being to survive, they either need to have advanced self-healing, like Tsunade, or have Nigh-Invulnerability like the Third Raikage.
Kimimaro's ability to generate vast quantities of bone and project it from his skin as weapons and/or armor would necessarily require him to have some kind of healing factor so he doesn't hemorrhage to death every time he uses it.
Kabuto's Sage Art: White Rage Technique acts as a flashbang combined with producing enough noise to make the air oscillate from the vibrations, blinding and paralyzing the opponent. He must use his snake-like cornea to block out the light and Suigetsu's Hydration Technique to liquefy his organs so they can absorb the sound.
In Claymore, when Clare awakens while fighting Rigaldo, she cannot control her speed and winds up running into more than a couple buildings.
Those Claymores who are known by the Organization to have shape-altering abilities are issued special uniforms that can stretch to accommodate the changes.
Durarara!!'s Shizuo acquired Super Strength pretty early on in life, but not the physical resistance that usually goes with it. The end result was that he spent as much time tossing cars as he did in the hospital from the resulting fractures until around high school.
Subverted with Zelgadiss in Slayers, a human/demon/rock golem chimera whose Nigh Invulnerability comes from him having skin made of stone (and hair that resembles steel wire.) Due to his "monstrous" (in his opinion) appearance, he's willing to lose his supernatural powers as long as he can become fully human again.
The various physical laws that Digimon break (size-shifting, thermodynamics, etc.) can be explained by the fact that their world is a big computer program. How they do it in the real world, however...
It is assumed that the superstrength of the vampires in Hellsing comes with a limited form of telekinesis to deal with recoil. It's the only way that Seras could fire a gun that weighs three times as much as she does on full auto without bracing against anything without going flying.
The fact that one enemy's force field was not air-permeable was the key to her defeat in Dragon Half. Mink breathed fire at her until all the air in the bubble ran out.
Being omnipotent without the omniscience to deduce how your changes will play out is a significant plot point in Serial Experiments Lain.
Chirico from Armored Trooper VOTOMS has a powerful Healing Factor but still feels pain the way a normal person would. As he result he has serious psychological problems due to suffering through the agony of countless things that should have killed him, like being cut into pieces or burned alive with high octane fuel.
In Mahou Sensei Negima!, the main downside of Negi's RaitenTaisou is that Negi doesn't have the required secondary powers, and must control himself through extremelly advanced wind magic, which, due to requiring him to previously "mark" the destination and making him unable to actually react to change course mid-air, make it very possible to counter it by just obstructing the way between Negi and his target. Raiten Sousou, its upgraded version, provides the required secondary powers.
Several of the heroes in Tiger & Bunny. The strong, and/or fast ones (Kotetsu, Barnaby, Antonio) are durable enough to use those skills without injuring themselves. Karina and Nathan are likely to have resistance to their own elements, and Pao-Lin explicitly states that a taser won't work on her. Possibly odd example in the case of Lunatic who can create fire in his hands for use in his crossbow, but his flashback in episode 16 shows that once he's set light to something (or in that instance someone) his fire can still harm Lunatic himself, as his father quite deliberately burnt young Yuri's face.
In High School D×D, Issei's power is having Boosted Gear, a Sacred Gear that doubles his power every ten seconds to the point that he can kill any deity whatsoever. The drawback however is that his stamina also gets drained because his body cannot handle the stress of his Sacred Gear's power. This leads him to undergo Training from Hell twice, once from Rias, and another one from Tanninim.
It's eventually explained that the whole reason Squid Girl can pick up people and other heavy objects with her tentacles without losing her balance is that she can alter her weight with her bracelets, becoming heavy enough to anchor herself.
At first one might think that the armors in Saint Seiya protect the users from the opponents' attacks; in fact, the armor's main function is to protect the user's body from his own powers since a normal human body isn't able to resist such amounts of power. Of course Fridge Logic kicks in most fights since more often than not the characters end up half-naked.
In one episode of Mega Man Star Force, Cancer Bubble grew to a prodigious size. However, as he didn't become any heavier or more powerful, he was still a pushover for Mega Man.
In Medaka Box, part of Mogana Kikaijima's training as a swimmer gives her superhuman lung capacity, allowing her to hold her breath for up to a month or use her voice as a weapon.
In Is This a Zombie?, Ayumu Aikawa is a zombie, meaning he can survive and recover from nearly any injury. Unfortunately, he can still feel pain. He also has super strength, but since his body is not super durable, he can break his own arm from the force of his own punch.
Get Backers: Himiko Kudou's claim to fame is using various types of perfumes for combat. These range from a degenerative perfume that regresses the mental capacities of the victim, a flame perfume that sets the air on someone's lungs on fire, and so on. Her "ultimate" perfume makes her able to attain greatly increased speed (multiplicatively, by the number she smells her perfume) by exerting more force to her muscles and reactivity. However, she does not have the required powers of infinite muscle/bone durability, and so her entire body nearly breaks down when she used it. She never did use it very much precisely because of this reason.
She does have a required secondary power for everything else: since her very power depends on people breathing her poisonous perfumes, she had developed a method of breathing by sensing which air is safe enough to breathe (which basically entails a heightened sense of air pressure reading). As proof, her own sweat and body odor is a mild hypnotic psychoactive "perfume". The fact that she herself isn't affected by it is proof of her secondary powers (or at least demonstrated tolerance to her own poisons).
Gokukoku No Brynhildr: Kotori's teleportation resets the kinetic energy of the target, which is how she can rescue someone from a multi-storey drop without them splattering on contact with the ground.