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.)

(Edit number 2:)  I see my original answer did not address the last part of the question.  For that you are requiring morphisms in all dimensions to be potentially non-(invertible up to higher cells) so there is a non-reversiblility about things. Chris's examples give some idea of this but there is a nice set of ideas that have not been fully explored as yet that may give another. The context in which this arises is that of directed homotopy. This arises is computer science when modelling concurrent and distributed computing. An action takes time and resources, so is non reversible. If you model things by a directed space (and there are various interpretations of that idea see Marco Grandis' book for instance), and then use directed $n$-simplices for all $n$ and the test spaces, you get a singular complex with quite `singular' properties! (Look at directed space in the nLab for some ideas of what is going on here.) I tried to capture some of this in a paper (Enriched categories and models for spaces of evolving states, Theoretical Computer Science, 405, (2008), pp. 88 - 100.) The strucutre would seem to be related to the Segal category types cosntructions, but an adequate description is still lacking. 

The challenge is then to solidify this link and then to findout if it does give an adequate model of the sorts of situation modelled by the directed spaces in the first instance.