But that uses associativity, and it's the associativity that fails. I believe that "a+b" is always equal to "b+a".

A more simplified version of what you say is that:

  1 + x + x + x + x != x + x + x + x + 1

when x is suitably small. But to convert one into the other requires both associativity and commutativity:

     (((1 + x) + x) + x) + x

  -> (((x + 1) + x) + x) + x using commutativity
  -> ((x + (1 + x)) + x) + x using associativity
  -> ((x + (x + 1)) + x) + x using commutativity
  -> (((x + x) + 1) + x) + x using associativity
  -> ((x + x) + (1 + x)) + x using associativity
  -> ((x + x) + (x + 1)) + x using commutativity
  -> (((x + x) + x) + 1) + x using associativity
  -> ((x + x) + x) + (1 + x) using associativity
  -> ((x + x) + x) + (x + 1) using commutativity
  -> (((x + x) + x) + x) + 1 using associativity

If you write Y as x+x+x+x, then 1+Y == Y+1.

It's the associativity that fails.