The category $Rel$ of sets and relations between them fails to have finite (co)limits.

So does the $(2,1)$-category $Span$ of sets and spans between them (the pushout of the forward map $2 \to 1$ against itself fails to exist).

These are examples illustrating the point that self-dual categories often fail to have nice categorical properties (Hilbert Spaces, as mentioned by Andre, would be another).