A scheme is a ringed space locally isomorphic to an affine scheme. An affine scheme is the spectrum of a commutative unital ring together with a structure sheaf.
What are examples of "interesting" affine schemes wich are not (finitely generated) $k$-algebras? Put differently, the definition of a scheme seems very reasonable, but why not take an affine scheme to be something more handy?
I'd be most interested in operations one can perform on schemes modelled on spectra of (f.g.) $k$-algebras whose result is a scheme not modelled on a (f.g.) $k$-algebra. In particular something starting with finitely generated $k$-algebras (truely geometric objects) and going to something which isn't even modelled on spectra of $k$-algebras would be cool.
A number theoretic example would only be "interesting" with an explanation why it is advantageous to consider it in the context of schemes. An advantage would be, for example, a scheme theoretic theorem which may be applied to both, the number theoretic and the geometric context, and yields non-trivial results in both cases.