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 http://www.springerlink.com/content/r8t12847357j1ln7/

Apparently it is also mentioned here: L. Berrone, "The Associativity of the Pythagorean Law", The American Mathematical Monthly, Vol. 116, No. 10, Dec., 2009 http://www.jstor.org/discover/10.2307/40391255?uid=3738232&uid=2129&uid=2&uid=70&uid=4&sid=21101035566283

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

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