Coming straight from the Official Collector's Edition Strategy Guide, here is a list of treasures (either dropped by monsters or found in the world), that you should not sell.
They are needed to either complete quests or to upgrade weapons.
- Metal Scrap
- Rusted Bits
- Dynamo
- Magitek Core
- Glass Gemstone
- Earth Gemstone
- Sky Gemstone
- Magnetron
- Monster Claw
- Cylinder
- Coreurl Whiskers
- Barbed Scythe
- Cactuar Needle
- Spiked Armor
- Sturdy Helixhorn
- Repair Kit
P.S. All other monster treasures are okay to sell for loot.
Note
The list of monster dropped items is a lot larger than that, but according to the guide they serve no purpose other than to get gill.
There are ingredients and treasures that would be a good idea to horde, to give big effects to spells but other than that it appears the list is rather short, for stuff to keep and don't sell.
Source: https://gameplay.tips/guides/2140-final-fantasy-xv.html
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