Let $K$ be an algebraic number field and let $c = c(K)$ denote the residue
at $s = 1$ of its zeta function. It is known Wikipedia: class number formula
that c can be determined via
$$c = \frac{2^r (2\pi)^shR} {w\sqrt{(D)}}. $$
Some specific (approximate) numerical values for low degree fields are : $c = .785$ for $\mathbb{Q}[\sqrt{-1}]$, $c= .430$ for $\mathbb{Q}[\sqrt5]$, $c = .814$ for $\mathbb{Q}[\sqrt[3]2]$, and of course $c = 1$ for $\mathbb{Q}$ itself.
Question: What range of values is taken on by $c(K)$ as $K$ varies over quadratic and cubic extensions of the rationals? In particular, what upper bounds are available?
The Brauer--Siegel Theorem addresses this. It says that, over families of number fields of fixed degree, that $\log (h_K R_K)$ is asymptotic to $\log \sqrt{D_K}$.
This paper seems to go into some details on explicit bounds, in the odd degree case. In the case of quadratic fields, I think the result is ineffective, due to the possibility of Siegel zeroes. (This wikipedia article also briefly discusses the $L$-function estimate the underlies the Brauer--Siegel result in the quadratic case.)