> When you say "every number n there is a cube with n on it",
  > what does "every" mean.

To be more precise, every finite number.

  > Does every number include infinity?

No, in these sorts of discussions infinity is never considered to be a number. You need explicitly to be discussing transfinite arithmetic, and we're not.

  > Or should not not consider infinity to be a number?
  > If the latter, this is probably where I went wrong.

Absolutely you should not be thinking of infinity as a number.