We construct examples of infinitely generated almost *-moduels, i.e. of infinitely generated small modules satisfying the Colpi's conditionC(K) for allR< N0. We also prove that the examples are shrinkable (in the sense of Facchini and Salce). Hence, they do not satisfyC(N0),and the are not*-modules. This implies that the possible solution of the open problem whether each *-module is finitely generated depends substantially on the validity ofC(R) for infinite cardinalsR