Infinity for Nietzsche in at least one line of argument involves the eternal return; he refers to it in the Die fröhliche Wissenschaft and Also sprach Zarathustra; most completely in his Notes on the Eternal Recurrence:
Whoever thou mayest be, beloved stranger, whom I meet here for the first time, avail thyself of this happy hour and of the stillness around us, and above us, and let me tell thee something of the thought which has suddenly risen before me like a star which would fain shed down its rays upon thee and every one, as befits the nature of light.
Fellow man! Your whole life, like a sandglass, will always be reversed and will ever run out again, - a long minute of time will elapse until all those conditions out of which you were evolved return in the wheel of the cosmic process. And then you will find every pain and every pleasure, every friend and every enemy, every hope and every error, every blade of grass and every ray of sunshine once more, and the whole fabric of things which make up your life. This ring in which you are but a grain will glitter afresh forever.
And in every one of these cycles of human life there will be one hour where, for the first time one man, and then many, will perceive the mighty thought of the eternal recurrence of all things:- and for mankind this is always the hour of Noon.
This thought is one echoed in Indian Metaphysics - the cyclical universe and in physics via Poincares reoccurence theorem which is traced to a question in Celestial Mechanics - the question of the stability of the Solar System.
But can repetition characterise infinity? Or should it be natality, that is true infinity is characterised by non-repetition that is however 'far out' one goes nothing repeats, there is always some modality, some aspect that is essentialy new?
In Spinozoan Metaphysics, for exampe, there are an infinite number of modes that are essentially different from each other; the first two being extension (ie matter) and thought - the incommensurability of the two is exactly the hard (ie very dificult and probaby impossible problem of consciousness); here Spinoza is implicitly remarking that the infinity (of God) is characterised by plenitude, by incommensurability and by fullness.