There is a relatively well understood class of deformable W-algebras,
resulting from Drinfeld-Sokolov (DS) type reductions of Kac-Moody alge
bras, which are Poisson bracket algebras based on finitely, freely gen
erated rings of differential polynomials in the classical limit. The p
urpose of this paper is to point out the existence of a second class o
f deformable W-algebras, which in the classical limit are Poisson brac
ket algebras carried by infinitely, nonfreely generated rings of diffe
rential polynomials. We present illustrative examples of coset constru
ctions, orbifold projections, as well as first class hamiltonian reduc
tions of DS type W-algebras leading to reduced algebras with such infi
nitely generated classical limit. We also show in examples that the re
duced quantum algebras are finitely generated due to quantum correctio
ns 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 ge
neral. As a by-product, we also explain the origin of the previously c
onstructed and so far unexplained deformable quantum W(2, 4, 6)- and W
(2, 3, 4, 5)-algebras.