2 Billion Enchants

In Minecraft versions below 1.17.1 enchants could go up to level -2,147,483,648 to level 2,147,483,647. This is also known as the 32-Bit integer limit.

Sharpness 1610605583
Deals 1024 damage. Needs more info.

Sharpness 2727997
Hitting anything with this sharpness level heals them for 1 HP

Miscellaneous Effects

 * Swapping between 2 sharpness 2 billion items would ignore Immunity-Frames
 * These seems to be a separate kind of damage from the damage sharpness deals, this kind of applies to all enchants larger then the 16-Bit integer limit (−32,768 to 32,767).