People make all sorts of challenges or speed runs and the like for any game they find. Fire Emblem is no exception. We the obvious challenges, like no stave healing run, and there are others you probably thought about and want to do. So, if you have any challenge run in mind, tell about it here in this thread.
I'll start us off:
No magic attack run: It's common with fire emblem that people will use at least one mage in their entire playthrough. Magic is especially helpful in defeating enemies with high defense. Therefore, not being able to use magic, or use a mage at all, means more difficulty against enemies like generals and various bosses.
No use of vulneraries and elixirs run: sometimes, a healer isn't close enough a unit to heal. Therefore a unit uses a vulnerary to heal theirselves before they die by a sudden attack. With no use of healing items means more difficulty in keeping units alive.