[chrispenycate]

To a mathematician, infinity is not merely uncountable, it has a meaning outside numerical theories. The difference between factorial googleplex, which is a very large number indeed, and infinity is infinite. If you had an infinite number of monkeys, and typewriters to match you would be certain to have the collected works of Shakespeare, and every other literary work that can be transcribed in the roman alphabet, each an infinite number of times, because there are a finite number of possible combinations and infinity divided by anything finite, however large, is always infinity. (We'll leave Cantorian transfinites till later, OK?)
If the big bang theory is correct, the universe, though unbounded (and thus etymologically speaking without limits ie, infinite) is actually finite. Very big indeed, but not mathematically infinite. So we've nowhere to put all those monkeys. If since the creation of the universe every particle in the universe has created a complete other universe very minimum transition time, we've still got an infinity of monkeys left over (and some of the ones in intergalactic space are getting pretty bolshy about typing).
A cubical planet; do you need one? The highest probability is of an artificial constuct, but I think I could construct one with cosmic strings (wouldn't last long; that's an awful lot of gravitational strain. And the corners would probably stick out of the atmosphere)
Did you want dots on the faces, too?