A Pokémon with the Hyper Cutter Ability cannot have its Attack Statistic lowered by another Pokémon's attack or the Intimidate Ability. However, the Attack Statistic can still be switched as a result of certain moves.
The only known effect outside of battle appears in the game Pokémon Emerald. When a Pokémon with the Hyper Cutter Ability uses Cut, twice the amount of tall grass gets cleared out compared to a Pokémon with a different Ability using Cut.
Pokémon With Hyper Cutter