An identity $E$ that obeys all the claimed properties is
$$ E: x+(y+z) = (x+y)+w$$
for all $x,y,z,w$.

 - $E$ is implied by triple constancy (and hence by constancy): obvious since both sides are constant in this case
 - $E$ does not imply triple constancy (and hence does not imply constancy either): follows from considering the [left-zero semigroups][1] $x+y=x$ mentioned by arsmath
 - $E$ implies associativity: obvious by specializing to $w = z$
 - $E$ is not implied by associativity: follows from considering (say) addition on the integers


  [1]: https://en.wikipedia.org/wiki/Null_semigroup#Left_zero_semigroup