The authors introduce a new notion of complexity \(p^*\) for an infinite sequence over a finite alphabet. Instead of counting all the factors (blocks) of given length occurring in the sequence, they count all the blocks of given length that occur infinitely often in the sequence. Accordingly they define *-Sturmian words. For a sequence indexed by \(\mathbb{N}\) these are exactly the sequences for which \(p^*(n)\leq n+1\) \(\forall n\geq 1\). The paper announces many interesting results.