  We look at a class of transcendental real numbers xi which, together with
their square, satisfy some extremal property of simultaneous approximation by
rational numbers with the same denominator. We give a sufficient condition for
such a real number to have bounded partial quotients and construct new examples
of extremal real numbers with this property. These include all real numbers
whose sequence of partial quotients coincide up to its first terms with the
limit of a Fibonacci sequence of words on the set of positive integers,
starting with two non-commuting words (so that the limit is not ultimately
periodic).
