Double-edged sword: As often as not, really amazing powers or techniques have some sort of handicap or restriction. Depending on the hero, genre, or show these can sometimes be overcome, making said ability even more amazing.
The serious laugher: A protagonist that acts silly, happy-go-lucky, careless, clumsy, innocent, weak, etc, but becomes super serious when a dangerous situation occurs. This usually goes hand in hand with "Awesome thing X," as said person generally has some amazing skill or trait.
The power of rage: A character's power is often proportional to the amount of anger he has at a given time. Villian downfall #1: say something to shock or anger a main character the extent that he or she completely loses it. Ironically villans often do this on purpose because of some twisted desire to have hero fight at their level, or they just enjoy watching suffering in the first place. This almost always results in said villain getting dominated.
--The stupidity clause: The power of rage can backfire if the person in question is hopelessy inexperienced or is fighting an enemy for the first time, charging uselessly without knowing the first thing about him.
Second time's the charm: A hero or main character suffering a major defeat is practically guaranteed to do a rematch with the one responsible for it and do a hell of a lot better than before, regardless of whether or not he actually succeeds.
"It's the new model!" The newest robot is always the best one and is likely to dominate the others until the next big thing comes along. This cliche/law can be overridden by Awesome thing X or the power of rage, depending on the circumstances.
--The Gundam clause: In a fight of the latest Gundam against the most powerful robot/mobile armor/state-of-the-art moving weapon of doom, the Gundam always wins.
Nature Over Tech: Using technology to enhance or override a naturally existing skill or trait is almost always a bad idea and very likely to create an inferior version of it or otherwise backfire. See: artificial newtypes (Zeta Gundam), ZERO System (Gundam Wing), engineered humans (Seed Destiny).
This also applies in a broader sense. Miyazaki's movies often highlight and praise the beauty of nature over industralization. I'm fairly convinced that there is no such thing as an anime that values industrial progress over natural beauty.