EarthBound/Cave of the Past
 Piece of Meteorite
Back at Saturn Valley you'll see that Dr. Andonuts and the Apple Kid have created the Phase Distorter II. Unfortunately it still doesn't work... You'll need a piece of Zexonyte which can be found in meteors. The only meteor we know about is the one back at Onett! Teleport there and head up the mountain to the meteor that started this entire adventure.
Be sure to watch out for the Ghost of Starman and the Evil Eye enemies, which can be extremely deadly. The Evil Eye can diamondize you! Remember that you can visit mom to rest up. Check the meteorite for Zexonyte at the top of the mountain and then teleport back to Saturn Valley.
Give the Zexonyte to Dr. Andonuts and then sleep in the hotel. The Mr. Saturns have new items in there shop so make sure to stock up before moving on. You are getting close the end of the game here. Once you feel you are ready head into the Phase Distorter. At this point there is no turning back!
 Cave of the Past
You'll find yourself in a strange cave that you may have seen earlier in the Lost Underworld. Follow the path until you encounter the Star Master, who'll teach Poo the ability PSI Starstorm Omega! If you check the rubble of the broken Phase Distorter you'll find a Horn of Life. After a bit Dr. Andonuts, Apple Kid, and a Mr. Saturn will appear in another Phase Distorter that can actually travel through time! Unfortunately, you'll have to be placed in robotic bodies for it to work without killing you.
Before you know it you land in the past as robots. The enemies here are the most difficult in the game. Keep in mind that the Phase Distorter can now replenish your HP and PP as well as call your dad to save the game. The worst enemy is the Final Starman which can not only use PSI Starstorm Omega, but also has the ability to revive fallen party members. Take him out first, always. Watch out for the Nuclear Reactor Robots which can explode, dealing massive damage to your team.
Whatever you do, do not use any items on these enemies. Also, do not heal yourself with items. You will need everything you brought with you to take down the final boss. If you come across a Bionic Kraken, it has a 1/128 chances to drop a Gutsy Bat, which is very powerful. However, as it is the end of the game, it is not worth attempting to find unless you are a perfectionist.
Follow the path and pick up the Legendary Bat along the way. This is the only item you'll find, so just work your way along the path. You'll eventually enter a large brain-like pathway. At the end of it, you'll encounter the final battles of the game, Pokey and Giygas.