I can’t think of any specific game but I can think of a few challenge modes I’ve beaten that were hard.
God Hand: Kick Me challenge. You can’t use any of the God Hand’s powers.
MGS 3: No Alert run.
Viewtiful Joe: Beating the game with Captain Blue. The danger warnings go away making dodging tough.
Metroid Other M: Beating Hard mode. There aren’t any health powerups/missile tanks.
Castlevania Portrait of Ruin: Hard Mode Level Cap 1.