With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known *for all* $i\geq 0$? 

Here, by "trivial examples" I mean for instance examples like $R = \mathbb{F}_q[t]$, or other examples that can easily be deduced from Quillen's computation of $K_i(\mathbb{F}_q)$, the zero ring, etc.

I believe there are no others but I'm not an expert (just curious) and I am having trouble deducing this from the literature.

For instance the $K$-groups of an algebraically closed field are divisible but (correct me if I'm wrong) the uniquely divisible part has not been determined.