Clones containing all almost unary functions

Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions. It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all such clones are finitely generated over the unary functions. In particular, we obtain an explicit description of the only maximal clone in this part of the clone lattice. This is especially interesting if X is countably infinite, in which case it is known that such a description cannot be obtained for the second maximal clone over the unary functions.