Can we glue (in this sense) finitely many integral Noetherian affine schemes whose underlying set is finite along open subschemes so that the glued scheme is separated and is not affine? Note that the schemes being glued are at most one-dimensional so open subschemes are automatically affine.
Some examples: if one glues the spectrum of $\mathbb{C}[[x]]$ to itself along the generic point, then one gets a non-separated scheme. If one glues the spectrum of $\mathbb{C}[x]_{(x)}$ to itself along the generic point with twisting $x\leftrightarrow \frac{1}{x}$, then one gets an affine scheme isomorphic to the spectrum of the intersection of $\mathbb{C}[x]_{(x)}$ and $\mathbb{C}[x-1]_{(x-1)}$ inside $\mathbb{C}(x)$.