One thing I've noticed is that addition and multiplication both form commutative groups over the reals, but subtraction, division, and exponentiation are neither associative nor commutative. Ignoring issues with closure for division and possibly exponentiation, all 5 have the property that $(a \star b) \star c = (a \star c) \star b$ (that I call "right associocommutativity" because the swapped operands are on the right). Both left and right associocommutativity are implied by the combination of associativity and commutativity. However, tetration ($\uparrow\uparrow$, repeated exponentiation) has neither left nor right associocommutativity.
Now, my questions: Is there a better name for this? What other operations that aren't both associative and commutative have this property? Why doesn't this work for tetration? Is there a similar property that all of these operations have?