Unlike a Functor, a Monad can change shape?

Question

I\'ve always enjoyed the following intuitive explanation of a monad\'s power relative to a functor: a monad can change shape; a functor cannot.

For example: length

Answer 1

Just because the monad pattern includes some particular instances that allow shape changes doesn't mean every instance can have shape changes. For example, there is only one "shape" available in the Identity monad:

newtype Identity a = Identity a
instance Monad Identity where
    return = Identity
    Identity a >>= f = f a

In fact, it's not clear to me that very many monads have meaningful "shape"s: for example, what does shape mean in the State, Reader, Writer, ST, STM, or IO monads?