A not-so-serious answer; hopefully what it lacks in depth it makes up for by being elementary.

Suppose we forget Pythagoras's theorem and define a binary operation on positive reals by sending $(a, b)$ to the length of the hypotenuse of the right-angled triangle with side lengths $a, b$ forming the right angle.

The associativity of this operation is trivial in three dimensions but not so in two.

I came across this here:
D. Bell, "Associative Binary Operations and the Pythagorean Theorem", The Mathematical Intelligencer, Vol. 33, No. 1 (2011), 92-95, DOI: [10.1007/s00283-010-9171-6](https://doi.org/10.1007/s00283-010-9171-6)


Apparently it is also mentioned here:
L. Berrone, "The Associativity of the Pythagorean Law", The American Mathematical Monthly, Vol. 116, No. 10, Dec., 2009 https://www.jstor.org/stable/40391255