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][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

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][2] of the various identities discussed on this page and on the [related question linked by the OP][3], where the ordering is from stronger identities to weaker ones:

[![enter image description here][4]][4]

It might be a suitable undergraduate research project to extend this diagram to cover other short identities for magmas. EDIT: It might be a suitable graduate research project to find a way to do this automatically using proof assistants and possibly also machine learning/AI tools. (Some further discussion of this latter possibility can be found [here][5], where I pose a concrete challenge of using such tools combined with human expert-hours to extend the above Hasse diagram to the five thousand or so other universal equational laws for magmas that involve at most four applications of the binary operation $+$.)


  [1]: https://en.wikipedia.org/wiki/Null_semigroup#Left_zero_semigroup
  [2]: https://en.wikipedia.org/wiki/Hasse_diagram
  [3]: https://mathoverflow.net/questions/450890/is-there-an-identity-between-the-commutative-identity-and-the-constant-identity?rq=1
  [4]: https://i.sstatic.net/1e6lu.png
  [5]: https://mathstodon.xyz/@tao/110736805384878353