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UPDATE: A few days after posting this video, I reached 730 damage:

   • Bug Fables--730 Damage Against Single...  

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I dealt 623 damage to a single target in one round. Previously, on version 1.1.2 of the game, my best was 520:

   • Bug Fables--520 Damage Against Single...  

The recent v1.2 update added features that allow even more damage to be dealt. This video was done on v1.2.0.1. For the most part, my strategy is similar to the one I used in my 520-damage video, so see the description of that video for details. Here are the biggest changes I made:

I use a Berry Smoothie partway through the damage round in order to fully restore my TP. This allows me to have a lower maximum TP, so I can allocate my rank-up bonuses to MP instead. The extra MP lets me equip the medals listed below.
I equip Last Wind on Vi. This allows her to take an extra action if she starts a round in danger (HP less than or equal to 4). It also allows her to take an extra action if she is revived in the danger range; this can be done with Kabbu's Pep Talk.
I equip Item Recycler on Kabbu. This gives him a chance to use an item without consuming it, so he can use the same item multiple times. Along with the extra actions from Last Wind, recycling Hustle Candies allows me to start the damage round with 13 actions.
I equip We Owe Ya, allowing a helper to deal extra damage. The helper also knocks the king down so that Chompy can hit him with Poison Bite. For this reason, I don't need Poison Needles, so I can save some MP.
I equip Royal Calling instead of Empower+. Royal Calling enables the use of Royal Decree, which gives every ally both a red up arrow (which increases attack) and a blue up arrow (which increases defense). Empower+ only gives red up arrows. The blue up arrows are useful for surviving the king's attacks. Additionally, Royal Decree can be used by anyone, but Empower+ can only be used by Leif.
After killing Vi and Leif with Abombination, I revive them with two Pep Talks rather than one Miracle Shake. Although this might seem like a waste of an action, it is actually better because it revives Vi in the danger range. This activates Last Attack and gives her an extra action because she has Last Wind. After I turn relay from Kabbu and Leif to Vi, she has 4 actions instead of only 3. Additionally, I don't need to spend an action to poison Vi and put her in danger. Instead, I poison Kabbu ahead of time with Danger Dish, then use Status Relay to transfer the poison to Vi after reviving her.

There are quite a few luck factors in this strategy. One of the biggest is Item Recycler. According to this wiki page (which cites the game's code), Item Recycler has a 30% chance of working whenever the equipped ally uses an item:

https://bugfables.miraheze.org/wiki/Item_Recycler

I have 10 Hustle Candies and want to use them 15 times. With my strategy, Vi uses 2 of them, and those are always consumed because she doesn't have Item Recycler. Kabbu uses the other 8, so he must use them 13 times. There is about a 27.6% chance of being able to do this, assuming the information on the wiki is correct. Usually, Item Recycler only activates 2 or 3 times, whereas I need it to activate 5 times. In this run, I was abnormally lucky and got 8 activations.

Many aspects of the king's behavior are also random. Sometimes he refuses to attack Vi and put her in danger, which is necessary for Last Attack and Last Wind. Other times he kills Vi or Leif. His artifacts can also get in the way.

Another luck factor is the helper that's selected. There are 8 helpers, and some are much better than others. Ideally, I would get Maki as a helper; he's able to deal 6 damage. In this run I had Zasp, who can deal 4 damage.

I don't show the sign in the Cave of Trials at the end because it records total damage, not single-target damage, and this wasn't my highest total damage. On a previous attempt, I dealt 616 damage to the king and 10 to one of the artifacts, for a total of 626. Since the king didn't summon an artifact in this run, my total was only 623 and was thus not recorded in the cave.

There are various ways to improve this score, depending on how much grinding you're willing to do. The simplest option is to use the exact same strategy, but with Maki instead of Zasp as the helper. For a while, I restarted whenever the helper wasn't Maki, but the grind was tiresome and I decided to settle for Zasp. With Maki, you could deal 625. There are also many possibilities involving reducing the number of Hustle Candies. This could allow for more damage (e.g. by inflicting burn on the king), but the probability of success decreases rapidly as the number of candies decreases. Using 8 candies 13 times has a 27.6% success rate; with 7, it's 11.8%; with 6, it's 3.9%; beyond that, it's less than 1%. Theoretically, you could reuse candies dozens of times and get a massive number of actions, but it's very unlikely unless some sort of RNG manipulation technique is found.