Is there an example of nonassociative arithmetic addition?

As explained in the comments by several contributors, in most standard definitions of mainstream algebraic structures and applications, addition is supposed or defined to be associative (even in near-rings, the first operation is supposed to form a group). It is a useful convention to follow but one can define and study algebras with one, two or more internal binary operations (and even-ary operations) without this restriction but it is wise to use other names.

The two references you found are related to a line of research called "genetic algebra" by some, also "evolution algebra". See MSC 2010 subject 17D92, 17DXX started mainly in the forties by Etherington.

One of the main ideas is the combination/fusion/recombination of two gametes two make one new organism (that's close to an addition, but is non-associative since the order in which your ancestors reproduce matters) as well as the idea of seing familiar numbers as projection of trees. Etherington's name is now associated with various families of trees or equivalently to certain kinds of parenthesizing schemes.

See the Wikipedia entry for basic references on Genetic Algebra.

You might also be interested in other non-classical algebraic structures such as (planar) ternary rings, k-loops, operads but also, even if it is mainly associative, Tropical Mathematics.

Concatenation can be considered a type of ‘addition’, but concatenation of morphemes is not associative, a counterexample to associativity being the fact that ‘un(peeled)’ and ‘(unpeel)ed’ are antonyms (which is what is behind the fact that ‘unpeeled’ is an auto-antonym).