You must have a character deal 10 billion damage to an enemy in one attack, or witness an enemy dealing 10 billion damage to one of your characters.
This is really hard to pull off, as it requires extremely high stats to get close to even the 1 billion mark. However, a DLC character makes this very easy.
The easiest way to do this is to purchase the "Action Prinny" DLC. This is for the character from the PSP game "Prinny: Can I Really Be the Hero?", not Prinny Kurtis. Prinny can learn the evility "Hell's Finest". This grants 50% evasion, but the character takes 10x damage.
To do this yourself, you'll need a maxed rank 40 weapon, and preferably a male character. A Cosmo Infinity or Baal Sword will do fine. Once you've maxed it out, give it to your strongest character, who should be at level 9999. Doing multiple level 9999 reincarnations will boost his/her stats further. That character should know Final Arts (fist) or Infinite Graves (sword), and should be boosted as high as possible. You will also want a character that knows Brave Heart (and have it maxed) and one with Speed Boost (if using a fist).
Have Prinny learn and equip Hell's Finest. It unlocks at level 125 for him.
Once you've got this set up, go into any item's world. As you run through the levels, look for any level that has an Invincibility block. Have a character grab the block, and stand in a panel-free square if possible. Turn all of the other panels the same color (without clearing them), and put the Invincibility block down. Make sure to leave one enemy behind, so as not to clear the level. If there's a bonus ATK or Ally Boost block around, save those as well.
Have your attacking character meet up with another character on Invincibility panels. Have your attacker test hitting with your special. If you can do 150M+ damage, you should be on track for doing 10 billion. Have your two characters use standard attacks on each other to build up your combo bonus (maxes at 256, granting +475% damage). Tennis Rackets make this very quick, as they grant +4 counter rate (maxes at 9). They can be stolen from Sports Pirates.
When you're ready to test again, bring out Sapphire and have her stand next to your attacker, granting a 20% bonus. Have someone cast Brave Heart and Speed Boost (if necessary) on the attacker. Bring out Prinny and put him next to your attacker. Hit him with the special, and you may surpass 10 billion damage.
When I did this, my attacker had 33 million ATK and SPD with the Cosmo Infinity. With boosts from other characters, this went up to around 55 million. With a combo of 12x (+105%), I was able to deal 10,400,000,000 damage to Prinny with Final Arts.
There is another method to do this, which has the enemy do the damage for you. Pass all the Stronger Enemy bills, then fight Baal again in the Land of Carnage. Place Prinny just outside Baal's movement range. Cast Brave Heart on Baal. If he uses his ranged attack, it should deal 10 billion damage to Prinny if it connects.