...? I believe I said the set of hyperreal numbers, which would contain the real numbers cross joined with the set of infinites and infinitesimals. Technically, the infinity that bounds the natural numbers would be any of those infinites. I can't really point to one specific infinity.

Infinity wouldn't be in the set of natural, rational, real, or even complex numbers. It acts as a boundary for all of those sets of numbers, but you could have a set that also includes infinity, in addition to those sets, making them the extended number set.

However, I want to point out an issue with what you said about those being different infinities... That isn't strictly true. Natural numbers, rational numbers, and powers of ten are the same level of infinite. Crazy as it is to imagine, the sets can be functionally mapped to each other. They have the same infinite of elements to them. It isn't until you add in those irrational numbers that the level of infinite increases. There is a higher order of infinite more numbers in the irrational numbers than in the rational numbers, so that infinite IS bigger.