Let $\mathfrak{C}$ be the category of ($\mathbb{Z}$)-graded-commutative rings. Does this category have limits in it?
I am particulary interested in power series rings over a field. Is there a reasonable way to view such a ring as a graded ring?
Let $R = k[x]$ be a graded ring. Let $I=(x)$ be a graded ideal. Is $\varprojlim R/I^n$ an element of $\mathfrak{C}$?
Are you asking how to put a grading on, say, the ring of p adic integers?
The limit you wrote down is isomorphic in the category of rings to the ring of power series over $k$. A grading is a decomposition into direct sums. But power series have infinitely many terms...This is a kind of vague explanation for why I think it isn't possible.