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 $x+y=x$ 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