  We introduce and study equivariant Hilbert series of ideals in polynomial
rings in countably many variables that are invariant under a suitable action of
a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing
functions. Our first main result states that these series are rational
functions in two variables. A key is to introduce also suitable submonoids of
$Inc(\mathbb{N})$ and to compare invariant filtrations induced by their
actions. Extending a result by Hillar and Sullivant, we show that any ideal
that is invariant under these submonoids admits a Gr\"obner basis consisting of
finitely many orbits. As our second main result we prove that the Krull
dimension and multiplicity of ideals in an invariant filtration grow eventually
linearly and exponentially, respectively, and we determine the terms that
dominate this growth.
