The obvious concrete example is any Kan complex considered as a weak infinity groupoid. If that is not concrete enough, take a space and its singular complex is a weak infinity category. If you want category as against groupoid, the homotopy coherent nerve of a simplicially enriched category $\mathcal{B}$, is another example (provided $\mathcal{B}$ is `locally Kan' i.e. fibrant.) Thus setting size issues aside, the category of topological spaces yields an infinity category. (Look up homotopy coherent nerve in the nLab if you need.  It is a very neat idea.)

(Edit: I should have started by asking what `concrete' means for you.)