Let D be a list of single digits, and let k be a positive integer. We construct an infinite sequence of positive integers by repeatedly appending, in order, one at a time, the digits from the list D to the integer k, in one of four ways: always on the left, always on the right, alternating and starting on the left, or alternating and starting on the right. In each of these four situations, we investigate, for various lists D, how to find infinitely many positive integers k such that every term of the sequence is composite.