  There is a relatively well understood class of deformable W-algebras,
resulting from Drinfeld-Sokolov (DS) type reductions of Kac-Moody algebras,
which are Poisson bracket algebras based on finitely, freely generated rings of
differential polynomials in the classical limit. The purpose of this paper is
to point out the existence of a second class of deformable W-algebras, which in
the classical limit are Poisson bracket algebras carried by infinitely,
nonfreely generated rings of differential polynomials. We present illustrative
examples of coset constructions, orbifold projections, as well as first class
Hamiltonian reductions of DS type W-algebras leading to reduced algebras with
such infinitely generated classical limit. We also show in examples that the
reduced quantum algebras are finitely generated due to quantum corrections
arising upon normal ordering the relations obeyed by the classical generators.
We apply invariant theory to describe the relations and to argue that classical
cosets are infinitely, nonfreely generated in general. As a by-product, we also
explain the origin of the previously constructed and so far unexplained
deformable quantum W(2,4,6) and W(2,3,4,5) algebras.
