I guess the question is meant to ask abut "classical concrete categories" only, those whose objects are sets with structure. For otherwise, the class of examples that we certainly care about is vast and uninteresting (as a class). 

For instance a groupoid has an initial object precisely if it is equivalent to the trivial groupoid. 

Any disjoint union of two categories has no initial object. 

Any poset without smallest elements has no initial object.