Are there any clear, accepted examples of operations that are appropriately defined as "addition" but are not associative? Although I can find references to abstract discussions of arithmetic systems with nonassociative addition behind paywalls (Non-Associative Arithmetics, I.M.H. Etherington 1949) or in obscure books (Arithmetic in a Number System with Completely Nonassociative Addition, Mary Bearden Williams 1958) that are not available to me, I cannot find any examples of such operations.
For comparison, a clear example of nonassociative multiplication is the cross product of vectors in three-dimensional Euclidean space.
Machine floating point arithmetic is sometimes posited as an example of nonassociative addition or multiplication, but this seems a rather crude example because the lack of associativity is due to implementation-specific rounding errors, not an intrinsic nonassociativity of adding or multiplying real numbers.