Common topological constructions which are neither initial nor final I've come across the notions of 'final' and 'initial' topologies, which are the extreme topologies on a set $X$ which make a given family of functions to or from $X$ continuous. 
There are many examples of these topologies, however I would like to request a list of importamt non-examples which are relatively common in either analysis, geometry or algebra. For instance, the compact-open topology in functional analysis and the metric topology in analysis seem like obvious examples (can either of these be realised as final or initial topologies for an obvious class of functions?), but, for example, the Zariski topology can be considered as an example of the initial topology.
These two concepts seem to unify a lot of the constructions in topology, so understanding the counterexamples seems fairly important to me. I would greatly appreciate any references to other common categorical topological constructions.
 A: Many definitions in topology are important because they correspond to limits and colimits in category theory.  Since the category of topological spaces is fibred over the category of sets, all limits and colimits in the category of topological spaces can be constructed according to the following recipe: take the appropriate limit in the category of sets and then take the coarsest (for limits) or finest (for colimits) topology on that set making all the appropriate functions continuous.  
The main topological construction I can think of that does not arise in this way is the exponential $Y^X$ or the space of continuous functions from $X$ to $Y$.  Such a space does not always exist, but when it does, it is constructed using the compact-open topology.  Unlike constructions involving initial and final topologies, the compact-open topology can not normally be explicitly defined using a basis of open sets constructed from open sets in the original spaces (in the way that, for example, the topology on $X\times Y$ is generated by the basis of open sets of the form $U\times V$).  Instead, it is generated by a subbase - a collection of open sets with no conditions imposed.
As Travis notes in the comments, any topology on a space can be constructed as a categorical limit in some way, but it might not always be possible to give a uniform description of what that limit is for a general construction such as the exponential.
The exponential on topological spaces has a well-known category-theoretic interpretation, and is extremely important for construction of path spaces, loop spaces and so on.
A: The hyperspace construction isn't initial or final as far as I know. 
$H(X)$ is defined as the set of non-empty closed sets with the topology generated by the subbase $[U] = \{A \in H(X): A \cap U \neq \emptyset\}$ and $\langle U \rangle = \{A \in H(X): A \subseteq U\}$, where $U$ can be any open non-empty subset of $X$. 
It is "categorical" in the category of compact Hausdorff spaces: a map $f: X \rightarrow Y$ induces a map $H(f): H(X) \rightarrow H(Y)$ by $H(f)(A) = f[A]$, using that images of compact sets are compact hence closed, etc. In that same category, the hyperspace construction commutes with inverse limits. Metric compacta have a metrisable hyperspace (the Hausdorff metric).
A: The Tychonoff product topology on $X=\prod_{i\in I}X_i$ is the weakest topology that makes each projection $p_i:X\to X_i$ continuous. The strongest is the discrete topology. The box topology is stronger than the Tychonoff and weaker than the discrete. If each $X_i$ is a metric space, the uniform topology on X lies between the box and the Tychonoff.  
Unfinished thought : Perhaps there are some "non-examples" concerning compact Hausdorff spaces. If a topology $T$ on a set $X$ is compact Hausdorff, then no strictly stronger topology $T'$ on $X$ can be compact. 
