An identity $E$ that obeys all the claimed properties is $$ E: x+(y+z) = (x+y)+w \hbox{ for all } x,y,z,w.$$
- $E$ 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
This candidate $E$ was located by pursuing the analysis in Pace's answer to isolate the form that $E$ had to take as much as possible, as described in the comments to that answer. With a little more effort, it should be possible to entirely classify (up to relabelings and symmetries) the full set of identities $E$ that answer the question.
Here is the Hasse diagram of the various identities discussed on this page, where the ordering is from stronger identities to weaker ones:
