Almost all Noetherian rings that occur naturally in algebraic geometry or number theory are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings.
The concept is named after Alexander Grothendieck.
Here is an example of a discrete valuation ring A of characteristic p>0 which is not a G-ring.
If k is any field of characteristic p with [k : kp] = ∞ and R = k[[x]] and A is the subring of power series Σaixi such that [kp(a0,a1,...) : kp] is finite then the formal fiber of A over the generic point is not geometrically regular so A is not a G-ring.
Here kp denotes the image of k under the Frobenius morphism a→ap.