Interesting to see the perspective of Americans here.
Fed up can have different meanings depending on the context. In this context it has the connotation of bored, although as Richard it can also have the connotation of anger (but not really in your example). I agree with Richard that in this example the meaning of the two sentences is basically the same.
Aren't you fed up of playing that game?
Aren't you fed up with playing that game)?
Both sound fine and natural to me. I would favour the construction with 'of' myself, but I could see myself saying both quite easily.