Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?
Every commutative ring is the directed colimit of its subrings that are finitely generated as $\mathbb{Z}$algebras. The Hilbert Basis Theorem implies that these subrings are noetherian. Actually this method is used in EGA IV, §8.9 to generalize some theorems from noetherian schemes to more general schemes. 

