Given a ring $I$ and an ideal $I$, consider the $I$-adic metric. Then the completion of $R$ wrt the $I$-adic topology is the (Hausdorff) completion of the metric space $R$, therefore it may be performed as the set of Cauchy sequences. The algebras in the increasing union are the subspaces of $\hat{R}$ of sequences with prescribed order of convergence, and they turn out to be quotients of $R[X_1,.. X_n]$ with $n$ the number of a certain system of generators of $I$.

EDIT: Given a ring $R$ that is an algebra over a base ring it is always a filtering union of finite type algebras. Take a system of generators of $R$ over the base ring. The family of finite subsets of this system provides a collection of finite type subalgebras of $R$ whose filtered union is $R$.

(Some considerations on completion via Cauchy sequences deleted.)