The category $Rel$ of sets and relations between them [fails to have finite (co)limits](https://math.stackexchange.com/questions/354779/the-category-set-seems-more-prominent-important-than-the-category-rel-why-is-th/870483#870483).

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).