Running maximum of a k-regular sequence

The k-regular sequences form a large class studied in number theory, combinatorics, and other parts of discrete mathematics. This class is known to be closed under many natural operations, such as term-by-term sum, product, running sum, and so forth, but it is not closed under running maximum. Proving the previously-known counterexample, involving the Stern sequence, required intricate arguments. In this note, we construct a significantly simpler example of a k-regular sequence whose running maximum is not k-regular.