Even if you only like to work with smooth complex projective varieties, it's still possible to run into non-noetherian spaces.

Let X be a smooth projective variety over C, where C is the complex numbers. Then the tangent bundle TX is a natural vector bundle on X, representing first-order tangent information. Viewing it as a scheme, it admits a map TX -> X. Another perspective on TX is that it represents the functor

Y -> Hom(Y x\_C C\[t\]/t^(2), X)

(Check: the C-points of this are just tangent vectors on X.)

Now it's natural to consider also higher order tangent information. This gives rise to the notions of bundles of m-jets, which similarly correspond to m-order tangent information. If X\_m is the bundle of m-jets over X, then it represents the functor

Y -> Hom(Y x\_C C\[t\]/t^(m+1), X).

Now there are natural maps X\_m -> X\_n for m > n, and as a result there's a natural way to take the limit of the objects lim\_{m -> infty} X\_m. The resulting object turns out to be a scheme called the arc scheme, and you can think of the points of the arc scheme as being infinitesimal arcs on X, with all orders of tangent information. The arc scheme also represents a functor: it represents

Y -> Hom(Y x\_C C\[\[t\]\], X),

where C\[\[t\]\] is the ring of formal power series over C. However, the arc scheme is basically never noetherian! You can check that the dimensions of the bundles of m-jets increase as m goes to infinity, and since the arc scheme admits maps to each X\_m, you might see why it won't be noetherian in general.