The lime slime does not gain levels but instead will consume potions stored in its inventory, it will use these during battles before attacking. Any potions it drinks will apply permanently, unless your party is defeated.
Not exactly: They will apply normally and the buff will be lost after one fight, but there's an additional +5 or so that's applied permanently per potion.
You don't have to fight to make it consume the potions, you can do it in the overworld by just drinking them and choosing the lime slime as the target. You still have to get into a fight and end it by either winning or losing afterwards to see the permanent result though.
