What achievements or unlocks are only available by playing on Hard mode?

the-binding-of-isaac-rebirth

According to the Rebirth wiki, which by no means claims absolute correctness but could be a nice indicator:

  • God Head
  • Lost Baby
  • Cute Baby
  • Crow Baby
  • Shadow Baby
  • Glass Baby
  • Wrapped Baby
  • Begotten Baby
  • Dead Baby
  • -0- Baby
  • Glitch Baby
  • Fighting Baby

All of them except for Godhead are for beating Mom's Heart in Hard Mode with various characters.

As for Godhead:

Godhead is unlocked by beating the Chest, Dark Room and Boss Rush with The Lost.

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