Mazinger Z and its sequels played with this trope:
In Great Mazinger, the final battle is won when Mazinger, Great Mazinger, Aphrodite A and Venus A combine their strongest attacks to bring Cool Airship Demonika down.
In UFO Robo Grendizer, Mazinger-Z, Great Mazinger and Grendizer combined all of their attacks in one manga scene to destroy a Vegan Star Ship. And in one of the movies, Grendizer and Great Mazinger combined their Chest Blaster attacks to destroy a Saucer Beast.
In the first episode of Mazinkaiser, Mazinger-Z and Great Mazinger used their Chest Blaster weapons in combination to melt a Mechanical Beast to slag.
Great Mazinger and Getter Robo have also combined their attacks in several movies.
The Super Robot Wars games have exploited this. When several of the Go Nagai mechas show up in the same game, they are given combination attacks. It is usually called Final Dynamic Special, and it is always impressive: .
Subverted in Magic Knight Rayearth. When the three main characters combine their color-coded magic, red for fire (duh), blue for water (obviously), and green for wind (of course) they're shown as distinct hues, but the end result is pure white. However, a similar attack is performed near the end of the second season, which is indeed a swirling beam of these three colors.
Sailor Planet Attack! Although, in this case, the rainbow effect comes mostly from the color-coded glowing auras around each Senshi.
The anime had another version, this one a one-off, Sailor Planet Power. Unlike the other version, this one lanched the attack straight upwards instead of at a specific target.
In Sailor Stars, Sailor Moon's upgrade to Eternal Sailor Moon is caused by all ten Sailor Senshi combining their powers. The upgraded locket is adorned with nine gems, the colour of each one representing one of the other Sailor Senshi.
Slightly more realistic, but still artistic license, in Metal Armor Dragonar: The three Dragonars wield weapons called "photon bazookas", each of which fires a different colored beam; when they combine, rather than turning rainbow-colored, they turn white...the only problem is that the colors used are red, blue, and yellow, the primary colors in painting; for light, green replaces yellow.
In painting, those three primary colors, red, blue and yellow mix to create brown.
But then again, yellow is a secondary color of light, composed of red and green light...
Slight different in Katekyo Hitman Reborn! where Tsuna's guardians represent a different weather type and each weather attribute has one of the colours of the rainbow attached to it as their 'Dying Will Flame'.
Bleach: The Cero Sincrético is an example of this. It is a fusion of 2 different Cero; when used by Pesche and Dondochakka, it was purple and yellow. Technically, it would change with the user.
The members of the Glittering Crux attempt this during the final episode of Star Driver against Samekh. It doesn't work.
In the Chinese Hu lu Xiong Di series, the seven protagonists represent a color of the rainbow, each with different abilities. The antagonists will often cleverly separate them, as every one of the protagonists have a unique weakness that has to be backed up by the rest. In every season climax, the protagonists realize the importance of unity and love, and they strike back against the villains. This was done albeit differently in the second series, as the seven protagonists combined into one single entity with all of their powers integrated into it before fighting the Big Bad, who promptly transforms into a One-Winged Angel.
While not a team, Magic: The Gathering has several cards that combine the powers of the five colors of magic to do something powerful. In keeping with the trope, we have:
Legacy Weapon, which can instakill any card in play, typically with no chance to bring them back.
Coalition Victory, which, if you have a creature and a land of each basic color, wins you the game. Conversely...
Door to Nothingness, which forces your opponent to lose for the low cost of two mana of each color.
Progenitus, which can kill a player at starting health in two hits and has "Protection from everything" (meaning it can't be targeted by spells or effects, it can't be blocked by creatures, and it can't be damaged).
On that note, many cards require 2 or more types of mana in order to play them.
Some cards have the 'Sunburst' ability, which makes them more powerful for each different type of mana spent to cast them.
In the DCU's Blackest Night event, one of the only ways to destroy a Black Lantern is to combine light from at least two of the seven emotional energies and aim at the ring. It's also said that each of the energies is a portion of the White Light of Life, which has aspects of all seven. Subverted when throwing multicolored blasts at Nekron only makes him stronger; in the end only true, unbroken white light saves the day.
Wait until you have kids before you watch it. At least then there'll be some justification.
In Simon Hawke's Wizard novels, Kira's sapphire, Wyrdrune's emerald, and Modred's ruby send beams of colored light at one another when they form the "living triangle" and unleash the runestones' power on the Dark Ones and their minions.
Subverted in The Lord of the Rings. Saruman tries to claim that he has transcended being Saruman the White and become Saruman of Many Colors. Gandalf points out that when lights of multiple colors are truly combined, they make white light - Saruman didn't combine anything, he broke it apart.
"He who breaks a thing to find out what it is has left the path of wisdom."
In The Black Prism, By Brent Weeks, During the Defence of Brightwater Wall Gavin Guile uses all of the colors under his control to make an enormous explosion that kills a huge swath of the opposing army, and may or may not have created the thought impossible White Luxin.
A quite literal example in Elizabeth C. Mock's Children of Man series. Centuries ago, light magic was splintered into its six component colors: red, blue, green, purple, orange, and yellow. Since then, no one can access all six colors (the max seems to be four). However, Faela Durante has linked her mind with those of Kade Hawthorn and Jair Rafferty, using a very rare variant of red magic (red usually heals the body, Faela can also heal the mind). Faela has red and yellow magic, Kade has orange and trace purple, and Jair has green and traces of blue. Together, they could theoretically do the impossible and access light magic.
Live Action TV
Super Sentai and Power Rangers have frequently used this (very literally in the case of JAKQ). Originally, the general theme was that no one team member could finish off a monster of the week alone, but as time went on the members became more powerful. That said, the 'combined team weapon' has been a solid presence throughout and is not likely to go away. Usually, said weapon is so big it requires the whole team to hold it up and fire it.
Power Rangers regularly gives the team a BFG with this function; either assembled out of each Ranger's individual weapons or charged by a combination of everyone's energies.
Also, the Megazords sort-of count. Usually, they're formed from the combination of several Zords that each have the color of the Ranger that controls it, making their combined Megazord forms quite colorful.
A straight example of the Trope would be Wild Force Megazord's Mega Roar, in which each component Zord fired a beam in each Ranger's color, combining them into a downright purty gi-normous beam of destruction. In the Grand Finale, the Rangers' Zords and all the lost Zords hidden throughout the world hit the Big Bad with one combined attack, the Ultra Roar.
Played with in Jersey Jack Pinball's The Wizard of Oz with the Horse of a Different Color. You must collect at least four horses to get the bonus, and the highest score multiplier comes from either having all of them the same color, or all of different colors.
Disgaea's Prism Rangers. Notable in that they aren't able to pull off this attack until they've successfully assembled a full team of 7, which doesn't happen 'till Disgaea 2.
Another Nippon Ichi game, Phantom Brave, has the rainbow slash. Oddly enough, it isn't the slash that turns rainbow.
The Handsome Men (AKA "The Punishing Rangers") from Killer7 are a subversion of this. Their names range from the obvious "Handsome Red" and "Handsome Blue" to secondary colors like "Handsome Brown" and even one "Handsome Dead".
The Mad Midget Five from God Hand are yet another parody.
In Four Swords Adventures, when the Hurricane Spin is used, each of the Links leave a trail of their respective colors. You can choose to have them group together to have them all do it together and leave a trail of colors.
Super Metroid's Hyper Beam is an interesting subversion. While it's actually stolen from an enemy, it will replace all of the other color-coded beams in your inventory, AND causes both you and itself to flash different colors.
Done pretty much out of nowhere in VVVVVV. It happens when you first enter the Secret Lab, which requires all trinkets to access.
Notably averted in the final battle of World of Warcraft: Cataclysm. Each of the four Aspects fires a color-coded beam to the Dragon Soul in preparation for their final attack... which ends being a plain yellow beam.
Super Robot Wars UX: Ryuubi, Sousou and Sonken's + Ryofu's Soul of Three Seasons Combination Attack and Four Sovereign Armour Attack.
In Kingdom of Loathing, elemental attacks get colored damage indicators (in a black-and-white game). For this reason, attacks that deal all five types of elemental damage are called "prismatic attacks", and usually originate from rainbow-shaded items.
Captain Planet, the Trope Namer. The theme song actually went "Earth! Fire! Wind! Water! Heart! Go Planet! By your powers combined, I am Captain Planet!"
The "Care Bear stare" in Care Bears is a variation on this; it doesn't need a specific number or combination of bears to work.
Rainbow Brite does it a bit differently: the colour-themed members are needed to be there, but they're Barrier Maidens (and two boys) and battery packs, while the title character uses the colours for her own powers.
The Care Bear Stare-esque attack at the end of My Little Pony: Friendship Is Magic's two-part pilot is a partial example; the multi-colored gems associated with the Elements of Harmony (red, orange, pink, purple, magenta, and blue) don't comprise all the colors of the rainbow, but they form a traditional six-colored rainbow when their powers combine.
Interestingly, this seems to be part of FiM's tremendous aversion to green. Try it! Count the ponies who are honestly, properly green. Not light minty greenish teal (I'm looking at you, Lyra), not a slightly lime yellow, actual green.
W.I.T.C.H. has it occur right at the beginning when the girls first receive their powers, but don't know how to use them properly on their own yet. After that, they get more creative than a single combined blast.
The heroes of BIONICLE, the Toa, each control an Element. When six different elements meet, they form into super-hard, crystalline Protodermis. This has been used to imprison two villains, while an unfocused team attack (the beams didn't meet) defeated another early in the story.