Noob question about mobs.

[Teiren]

There actually is almost exactly what you're talking about. There are special, very rare, named monsters that appear in random spots in each zone of the game. These are part of something called The Hunt. They reward Allied Seals in ARR areas and Centurio Seals in Heavensward areas, and these seals can be traded for gear. But those monsters are only for Level 50 and 60, respectively, and they really are rare. You'll almost never see them unless you're looking in every corner of a zone, so when you're able to access The Hunt stuff, I'd just stick to the daily and weekly Hunt mark bills if you want the gear that comes from those seals.

There are also the Odin and Behemoth special boss FATEs in ARR areas which don't show up on the map and only occur every few days. Heavensward's equivalents are some kind of Coeurl and Proto Ultima FATE. Like the Hunt marks, this is only for Level 50 and 60 respectively. Nowadays, practically no one is doing them, and they reward tokens traded only for gear to be used for glamour (system that changes your gear's looks to whatever you want), but hey, it's similar to what you're asking about.

Btw, it's still worth rolling Greed on gear in dungeons you can't use. You can trade any gear with pink (Aetherial), green (Uncommon), or blue (Rare) text color to your Grand Company for Seals. Your most common source of gear as you level is going to be the Main Scenario and Job quests and NPC vendors, so make sure to check an armor vendor every time you arrive in a new town.