For a very explicit example, consider the ring $$k[x, y, x/y, x/y^2, x/y^3, \dots]$$ localized at the origin. This has value group $\mathbb{Z} \oplus \mathbb{Z}$ with lexicographic ordering (in other words, the $x$-value always is more important than the $y$-value).

It's easy to see it's not Noetherian but it does have finite Krull dimension, equal to $2$.

You can obtain this example geometrically, and explicitly, by repeated blowings up of the origin. See Hartshorne Chapter II, Exercise 4.12.

For a very explicit example, consider the ring $$k[x, y, x/y, x/y^2, x/y^3, \dots]$$ localized at the origin (ie, localize at the maximal ideal $\langle x, y, x/y,x/y^2, \ldots \rangle$. This has value group $\mathbb{Z} \oplus \mathbb{Z}$ with lexicographic ordering (in other words, the $x$-value always is more important than the $y$-value).

It's easy to see it's not Noetherian but it does have finite Krull dimension, equal to $2$.

You can obtain this example geometrically, and explicitly, by repeated blowings up of the origin. See Hartshorne Chapter II, Exercise 4.12.