Bucky

[npc=61160] (1,1,x) - s/s (ninja)
[npc=55356] (1,1,2)
[npc=77221] - (1,2,x) - p/p (destruction)

All pets used are rare and level 25; where applicable, breeds are listed.

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• Start with [npc=61160]
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• Use any attack or pass - either way, [npc=55356] is force-swapped in
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• [pet-ability=646]
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• [pet-ability=209]
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• [npc=55356] is force-swapped for [npc=77221]
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• [pet-ability=459]
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• [pet-ability=459]
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• Spam [pet-ability=640] until [npc=77221] is swapped for [npc=61160]
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• If [npc=77221] dies before forced swap, bring [npc=55356] back in and fill with [pet-ability=777] until [npc=61160] is pulled in
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• [pet-ability=389]
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• [pet-ability=384] until [npc=61160] is force-swapped for [npc=55356]
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• [pet-ability=209]
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This is a full series of turns to account for worst-case scenario, however, many times the full series is not needed to kill Bucky.

Alternate pet options:
Using the same strategy provided above, below are some possible alternate pets.

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• Potential alternates for [npc=61160] can be found here
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• [npc=71015] is a potential alternate for [npc=55356] if you use [pet-ability=116] in replacement of [pet-ability=777]
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• [npc=117340] is a potential alternate for [npc=77221]
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I was able to defeat Bucky first try with:

Clockwork Gnome (1,1,1)
Darkmoon Zeppelin (1,1,2)
Tranquil Mechanical Yeti (1,1,2)

Pet 1 gets swapped out in first round
Bombing Run then Missile with Zep until swap...after the swap, the bomb hits Bucky
Call Lightning with the Yeti, then Ion Cannon...Yeti will get swapped for Clockwork Gnome
Either punch him in the face (at this point, he was below 300 hp) with Metal Fist or Build Turret
Bucky dies

This works because it exploits the fact that Bucky is a beast.
The Call Lightning spell adds damage to your hits and gives a 25% damage increase for it's duration.
Hope this helps...cheers!

Humanoid team: (RNG dependant)

Anubisath Idol 2/2/1
Crusher 1/1/2
Ore Eater 1/1/2

Start off with deflection, stoneskin up to reduce the stampede and use demolish until deflection comes up, then use deflection. Swap to second pet.
Acid touch, Demolish until Anub idol is swapped back in by the portal attack.
Stoneskin, Demolish until stampede is about to end, deflect the portal attack, swap in second pet again. If second pet dies swap third pet in until the portal attack is used so anub is put back in automatically before he stampedes again.
Repeat the stoneskin, demolish until stampede ends and deflect of portal attack, swapping to other pet after deflection of portal attack if the debuff from stampede is still up.

The trick is to have the idol attack when stampede is up with stoneskin to reduce stampede to a mere 10 damage per hit, deflecting the portal, having any other pet soak the rushes until it decides to portal again after which it stampedes. With some minor luck on the Demolish, or maybe using another pet to increase hitchance, it should go down fairly fast.

Good luck.