Two other kinds of colimits:

1. Direct limits of schemes fail to exist. A good example is the following: let X be a scheme and Z a closed subscheme defined by an ideal I. Then for any n we get the nth infinitesmal neighborhood Z^(n) defined by the ideal I^(n+1) and a diagram $Z \to Z^2 \to \cdots \to Z^n \to \cdots$ and in general the direct limit of this diagram does not exist in the category of schemes. (It does exist however in the category of formal schemes). Knutson's Algebraic Spaces, Chapter 5, section 1 (near the end) explains this well (for algebraic spaces, but for this point it is fine to think of everything as a scheme). The point is that, while you can

2. Pushouts don't always exist in the category of schemes. The paper cited by Greg in the question you cited gives a pretty down to earth example (3.3) of the failure of a pushout to exist in the category of schemes.

Direct limits of schemes fail to exist. A good example is the following: let X be a scheme and Z a closed subscheme defined by an ideal I. Then for any n we get the nth infinitesmal neighborhood Z^(n) defined by the ideal I^(n+1) and a diagram $Z \to Z^2 \to \cdots \to Z^n \to \cdots$ and in general the direct limit of this diagram does not exist in the category of schemes. (It does exist however in the category of formal schemes). Knutson's Algebraic Spaces, Chapter 5, section 1 (near the end) explains this well (for algebraic spaces, but for this point it is fine to think of everything as a scheme).

A simple example is P^1 over a dvr (R,t), with Z the closed subscheme of P^1 defined by t.