We work in the category of **algebraic varieties** over
some algebraically closed field $k$.

By **infinite dimensional variety** I mean a filtration:
$$
V_0\subset V_1\subset V_2\subset\ldots
$$
where each $V_i$ is a closed subvariety of $V_{i+1}$.

For any affine variety X (reduced and irreducible), is it possible to give its coordinate ring $k[X]$ a structure of infinite dimensional variety in some canonical way?

This is trivially true for $X=\mathbb{A}^n$. More generally, I was trying to use the Noether's normalization lemma or look at the coordinate ring locally at some smooth point, but with no luck.