TITLE: Is there a finitely complete category with terminal object but NO subobject classifier?
QUESTION [5 upvotes]: This came up today while thinking about topoi in seminar, as the title suggests my question is;

Is there a finitely complete category with terminal object but NO subobject classifier?

Hopefully if the answer is yes, you can give an interesting example that is useful in some way. One of my professors has a saying that 

for every definition you should have an interesting example and an interesting non-example

So I am trying to stick to this. 
Thanks!

REPLY [3 votes]: I'm terribly late to this party, but the topic came up at the nForum, and so I might add some more observations. 
In any category with a subobject classifier, all monomorphisms are regular monomorphisms, which has some further consequences: the category is balanced (monic epis are isos), any two epi-mono factorizations of a morphism are canonically isomorphic, etc. Right away this kills off many examples such as the category of posets, the category of topological spaces, the category of commutative monoids, the (1-) category of small categories, etc. 
But what I think is even more of a killer is that in such a category, for any object $X$ the poset of subobjects $Sub(X)$ is a cartesian closed poset, so in particular has the property that meets distribute over any joins that exist (finitary or infinitary). So $Sub(X)$ is a distributive lattice (more specifically a Heyting algebra), provided that joins exist (which is usually true for categories of structured sets "in the wild"). "Most" categories do not have distributive subobject lattices (and it's usually not a hard condition to check, using the familiar "forbidden sublattice" criterion). For example, this kills off the category of groups, since subgroup lattices usually are not distributive; the same is true for abelian groups, rings, and so on. 
Details have been written up in the nLab article subobject classifier.