Enchantment over-leveling in survival

The maximum enchantment level possible in survival is V, IV for protection enchantments, feather falling, and piercing, III for most other enchantments. However, I would love to see higher levels than this, like X or XVI, etc. Being able to obtain them through trading and/or the enchanting table is obviously out of the question, and therefore could only be obtained through mob and chest loot. This should add an element of luck and encourage players to explore.

Side note: no players should be able to get it within the first in-game year.

Another side note: some enchantments are capped at a certain level.