[dfox]

On practical computers implementing such thing is trivial enough to be borderline uninteresting (at least when done by low-level non-portable means).

The interesting "practical" application is in proving that such a thing can exist in given formal system. By the way the concept of fixed-point combinators (of which Y-combinator is particular implementation) is essentually the same thing. (And in fact such combinators are notionally better match to problem "produce string that contains result of this function applied to it in its contents" than quines)

[Edit: functional->fixed-point and reworded the Y-combinator remark slightly]