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Only Squall needs to avoid Exp. Other party members are fair game. To avoid Exp Squall can be KO'd, or you can use Card or Break to safely dispose of enemies. You may also run, but it is important to stress that you must not deal damage to an enemy before running away because you will earn partial experience. If you accidentally kill an enemy because you were trying to lower their HP to Card them, or deal damage to something you can't Card or Break, you're basically forced to KO Squall before running or finishing the fight.

Besides Carding, Breaking, or KOing Squall/Laguna, a final option is LV Down from the Tonberry GF. Killing Tonberries for this GF awards NO exp, and can be earned as soon as you can pilot Balamb Garden. Using LV Down until it no longer works and then killing the enemy will grant a very minimal amount of experience, often in the single digits. This is occasionally important for some humanoids who are both immune to Break and Card.

Break spells can be earned relatively early in the game by winning Cockatrice cards, refining them into Pinions with Card Mod, and refining Pinions into Break's with Status Magic RF. Card and Card Mod can also be earned relatively early by killing Zell and Squall during the Dollet SeeD exam and grinding Soldier encounters with Seifer for AP

Additionally, keep in mind any Exp earned by Laguna will be transferred to Squall, so treat Laguna as you would treat Squall. Boss fights do NOT give exp with the exception of Odin for some silly reason. Scripted non-boss fights DO give exp AND can't be run from, so keep that in mind too. Save often in case you mess up because you probably will sooner or later.