Two off the top of my head are the "Cheapskate" challenge, and the "Sticky Materia" challenge.
Cheapskate: You can't voluntarily spend any money. This goes for items, inns, whatever. There are a couple of exceptions, where you are forced to spend the money to proceed. This is allowed under the rules.
There is a variation of this, in that whatever you sell is allowed to be spent before you exit the shop dialog. EG: You go in and sell 5 Ethers. You now have (IIRC) 5000 Gil to play with BEFORE you resume your adventure. You can't spend over that, and if you exit, that money locks. Think of it as a barter-type system.
Sticky Materia: The title speaks for itself. This one has three difficulty levels.
EASY: Once you equip any materia to a weapon/armour slot, it stays there. When you recruit someone that has pre-existing Materia, you are allowed the option to remove their Materia prior to their usage. You can equip weapons/armour that have reduced slots in order to remove materia. (If you have an all-cure set up, and you remove the cure, but not the all, that's bad luck.)
MEDIUM: Easy mode rules, but you can't change weapons/armour to remove materia. Any new items must have an equal or greater number of materia slots than the item you removed. Should you recruit someone with pre-equipped Materia, it must stay with them in its respective slot.
HARD: Same as Medium, but you must take the Wutai side-quest, and stick with the results. On the other difficulties, you are allowed a reset at this point.