We could have a lot of things. The thing is, L2 doesn't *start* off as an Ice Dungeon. It's a volcanic region.
We *COULD* have a snowy mountain area, but I wouldn't see where to put it. The Southeastern portion remains a volcanic region for the duration of the quest. :p
However, I may have a nice little spot a few screens to the east of L1, along roughly the northern central area of the map. It's mostly plains, even right up to the very top screen, but if we can get people into a tight spot, I can certainly see a combination of snow and grass over there, assuming the player has no other way to enter. (It'd be done via scrolling warps to trigger the Palette shift.)
