Chrono Trigger/The Masamune!
The Masamune! (600 A.D. third time)
Enemies: Gremlin, Bat
Items: Masamune, Taban Helm
Now that you're back at the end of time, make sure to fight Spekkio if you've reached another level set, i.e. 1-9, 10-19, 20-29, etc. You'll actually have one stop before you go to 600 A.D., you'll need to go to 1000 A.D Medina Village and give the sword to Melchior. If Lucca is in your party she'll help him reforge it, otherwise Robo will fill in. Once you have the Masamune in your possession, head back to the End of Time and warp to 600 A.D. to give the Masamune to Frog. Incidentally, at this point in the game, you still won't be able to go inside Guardia Castle, so there's not much else you can do in the present. Gaspar will also comment on the Masamune if you talk to him.
Head to the Cursed Woods to present the sword to Frog, and you'll witness a flashback explaining some of his past. He'll leave with you to go challenge Magus. One thing that is absolutely vital at this point is to take Frog back to the End of Time to learn magic from Spekkio. If you try casting a spell with Frog in the party when he doesn't know magic he'll comment on it briefly after the battle. Without it, he'll only be able to use two techniques total, and you'll waste all the time he could be learning water spells. So, backtrack to the mountain range, take Frog to Spekkio, and come back once he knows water magic. You should also go back to 65 million BC to equip Frog with better gear, as his armor is useless now. Also go to Lucca's House and talk with Taban to receive the Taban Helm for Lucca.
Anyway, now that Frog's fortified with magic, head to the Magic Cave, which is to the east of Fiona's Villa in the Middle Ages. You'll witness more of Frog's history, and watch him cut open the rock. Head through the magic cave, fighting monsters as you go or avoiding them. Take note of the sealed chest, and the little bat following you. You'll exit the cave on an island near Magus's castle. Step up to the gate, and welcome to the next chapter.
 Challenging Lavos
There are actually two endings you can obtain in this chapter depending on whether you challenge Lavos before or after Frog joins your party.
If you challenge Lavos before Frog joins your party: you'll get the People of the Times Ending, basically a bunch of random scenes involving different times and places.
If you challenge Lavos after Frog joins your party: you'll get The Frog vs Magus Ending. Frog will duel it out with Magus and his henchmen without your help, and the two will finish things off with a battle at the top of the castle.