[nimih]

Binary trees, with the operation being to join two trees at the root, form a magma which should quite clearly be non-associative (and thus not a semigroup).

There are of course other examples of sets equipped with non-associative binary operations (for example, 3-dimensional vectors with the cross product), but those typically have extra structure that you will want to exploit which "non-associative magma" cannot capture, and talking about them as magmas will probably feel somewhat artificial unless you have a good reason for ignoring all that structure.