  Iterated hairpin completion is an operation on formal languages that is
inspired by the hairpin formation in DNA biochemistry. Iterated hairpin
completion of a word (or more precisely a singleton language) is always a
context-sensitive language and for some words it is known to be
non-context-free. However, it is unknown whether regularity of iterated hairpin
completion of a given word is decidable. Also the question whether iterated
hairpin completion of a word can be context-free but not regular was asked in
literature. In this paper we investigate iterated hairpin completions of
non-crossing words and, within this setting, we are able to answer both
questions. For non-crossing words we prove that the regularity of iterated
hairpin completions is decidable and that if iterated hairpin completion of a
non-crossing word is not regular, then it is not context-free either.
