Directly east of Junon is a small cave that is only accessible by using Highwind or riding on a Blue/Gold Chocobo. Inside is a sleeping man who mumbles out your statistics when you speak to him. For example, he can tell you how many times you've fough and how many times you've escaped from battle. He also hints at the concept of "Master Materia".
The sleeping man gives you an item under the following circumstance: You must talk to him when the last two digits of the number of battles you've fough match. For instance, speaking to him after fighting 588 times may get you a Bolt Ring. You can only do this once.
However, there's one very special item you should look for - Mythril. Speak to the man who the last two digits match, and he may give you a piece of Mythril. You can then exchange this item with the weapon seller east of Gongaga Village for Aeris' Great Gospel Limit Break.