(The title was initially "What are your favorite concrete examples that you would calculate on the table during lunch to convince a working mathematician that the notions of limits and colimits are not as dreadful as they might appear?" but it was too long.)

This is a pretty basic question which may be more appropriate for another website. However, the examples I am looking for should appeal to a working mathematician. That is why I am asking here. 

Tomorrow I shall have lunch with a mathematician whom notions of limits and colimits make nervous in general. He feels that calculating small examples may help him overcome his fear. More precisely, what he is asking for are examples of various small diagrams (including fancy ones, for instance with loops) in familiar categories (topological spaces, abelian groups, &c.) whose limit or colimit we could calculate together over lunch, in the hope that he would get a better understanding of what are limits and colimits when they are taken over diagrams other than those giving pullbacks or pushouts (of which he already has a feeling). 

>Do you know of some particular instances of diagrams, in categories familiar to the working mathematician, whose calculation of the limit or colimit seems particularly illuminating? Or at least which could help a nervous mathematician overcome his fear of general limits and colimits? 

I could come up with ad hoc examples, but perhaps there is better than that?