After you make it through the Cave of the Past, you and your friends face the terrifying evil that is Giygas.
Have Paula set up a PSI Shield Σ to make Giygas's PSI attacks disappear. Giygas will use PSI Rockin α and PSI Rockin β.
Heavily Armed Pokey is vulnerable to Brainshock, so use that on him to make him feel strange. Even if his attacks hit Giygas, the deflection from Giygas's shield will only be 50% as strong as it would have been if Pokey attacked you directly.
Heavily Armed Pokey doesn't have any particular weaknesses to PSI, so try Freeze spells. Otherwise, use Multi Bottle Rockets and the Heavy Bazooka. Ness can attempt to use Paralysis on Heavily Armed Pokey for a 50/50 chance of paralyzing him.
After taking a lot of damage, Pokey will turn off the Devil's Machine, which contained Giygas's evilness. Then Pokey will disappear, leaving you face to face with the terrifying red swirl that is Giygas.
In phase 2, Giygas is weak to Brainshock, so use that. Giygas will cast a couple of incomprehensible attacks, which are in fact PSI Thunder β, PSI Freeze β, and PSI Flash Ω. Hopefully one of you has the Franklin Badge, and hopefully you're wearing Sea Pendants, Star Pendants, Night Pendants, or Earth Pendants to protect against Flash.
Don't use Multi Bottle Rockets on Giygas, because they don't work. The Heavy Bazooka, however, does work. Ness can use PSI Rockin Ω, which doesn't always work, but it's the best move to use, nonetheless.
After you've taken Giygas's HP down enough, Giygas enters phase 3. Pokey will appear and taunt you. This means that Giygas is almost done for. However, you can't hurt Giygas with your attacks now. The only thing that will hurt Gigyas now is for Paula to use the Pray command. You have to keep Paula alive to Pray nine times. After the ninth prayer, Giygas will die.
Focus your efforts on keeping Paula alive. It's a good idea to put the Franklin Badge on her so she'll be invulnerable to Giygas's thunder attacks. It will also help if you give her protection from Flash with a Night Pendant, Sea Pendant, or Star Pendant. If you don't have any of those, you can go for the Earth Pendant, although it doesn't afford 100% protection from Flash like the other pendants listed above.
Your success at this stage depends on your ability to keep Paula alive, so don't let the others do anything but defend themselves or heal Paula. Make sure that at least one of the others stays alive (Ness is probably best) to provide support for Paula. And don't forget that unconscious characters can still use the "Give" command to transfer their goods to the characters who are still in battle. This way you can still use food and items even if the character holding them is unconscious.
Congratulations! You defeated Giygas and saved the universe! Enjoy the awesome ending.