We're now entering the third Bracket, Onion Knight! Our competitors today are...
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Leviathan
Debuting as a dungeon of all things way back in Final Fantasy II, he finally became a boss and Summon in the follow up FFIII. Ever since, Leviathan has been a go to summon and one of Square's favorite subjects to show off the 3D graphics engine since this baby has been the first one designed for both FFVII and VIII's original demos and more recently in the rebranding trailer for FFXV. Leviathan's most noteworthy appearances have been as King of the Summoned Beast in FFIV who attacked Cecil's boat to get to Rydia, as a sealed beast from FFV who laid waste to one of ExDeath's fiends before challenging Bartz and company to a battle, and as the sealed Eidolon Garnet acquires to try and protect her mother from Kuja.
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Quetzalli
A summon obtained on the Solitary Island in a ruined world, Quetzalli (Palidor in the original Woosely script) taught the party useful time magic like Haste, Slow, and their upgraded Hastega and Slowga. In battle, this summon would swoop down and carry the party high in the sky, protecting them from enemy attacks before letting the party swoop down dragoon style to lay the hurt on their foes.
Our next competitors are...
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Alexander
Debuting in FFVI, and based off an early Amano design for the Bab-il Tower, Alexander quickly rose up in popularity and has appeared in most FFs since it's debut. Whether it's firing a Macross Missile swarm of Holy energy, or laying everything to waste with it's Holy Judgment attack, Alexander is the premier Holy based summon in the series. He was recently the star of an event in FFXIV, where he served as a dungeon and Primal for the Goblin races, waged a war long ago in XI's Van'diel, and most famously, battled Bahamut to a standstill while protecting the city of Alexandra in IX.
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Quezacotl
Debuting in FFVIII as the stand in for Ramuh, Quezacotl was one of the first summons most players got to see in VIII, belonging to Squall and used to help him pass his SeeD qualification test by felling Ifirit. He has only made minor appearances since then in trading card games, Record Keeper, Mobius Final Fantasy, and Revenant Wings. His Thunderstorm attack was pretty awesome to see as well.
Our next competitors are...
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Valefor
Debuting in FFX as Yuna's signature Aeon, Valefor works as a sort of mini-Bahamut for the game utilizing a strong non-elemental breath attack called Energy Ray or even it's souped up version you can acquire called Energy Blast. Valefor reappeared in FFX-2 as a boss for the Gullwings to deal with, briefly appeared in the Nautilus play about the War of Transgressions in FFXIII, and has made appearances in Dissidia Duodecim, Record Keeper and a brief mention in FXIV.
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Catoblepas
Debuting as a regular enemy as far back as FFIII, Catoblepas made the jump to being a summon in FFV, where he can be found in Galuf's homewarld, wandering the forests and turning people to stone with it's Demon Eye. He appeared again in FFVI as an esper that taught Bio, Break, and Death making him invaluable as being the second half of the Vanish/Doom trick. He has not appeared much since then, returning to his roots as a regualr monster in the series, including the upcoming FFXV, where he appears as an almost Admantoise size enemy on the field.
Our final battle for this bracket is...
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Cactuar
Debuting in FFVI as a desert dwelling enemy, the little mascot worthy enemy jumped up into the summoners league in VIII, where Squall has to defeat a Jumbo Cactuar to earn this critter. Cactuar has appeared as a summon in retro for FFVI and as a special move Zack Fair can use in Crisis Core. Pelting enemies with it's powerful 10,000 Needles, the critter has been a fun addition to the series.
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Zalera
Debuting in FFTactics as the Lucavi of Elmdore, Zalera made his summoning debut in FFXII where he tests people in the depths of the Baraheim Passage to defeat him and his army of the undead within five minutes. His signature move is Condmenation which uses the scream of the captive shaman he holds to instantly kills weak enemies and do non-elemental damage to stronger one. He makes other appearances in the Ivalice series as a scion in FFTactics A2 and as a tier three summon in Revenant Wings who teaches Basch his Apocalypse Shield ability.