Comparing infinite numbers Suppose you have 2 infinite numbers, say $A$ and $B$.
$A$ is an element of the hyperreals, so that $A$ is greater than every real number.
$B$ is the size of the set of natural numbers, $\aleph_0$
Does it make sense to compare $A$ and $B$? And if so, how can you compare these kind of numbers?

In a nutshell: No. Hyperreal numbers are "non-sets objects" while $\aleph_0$ is essentially a notion of size for a set. While hyperreal numbers can be represented as sets (but not only, e.g. in ZF+Atoms the atoms may be given the structure of the hyperreal numbers), they are interpreted as something else, while $\aleph_0$ is a lot more "concrete" as it will always be interpreted as a set of some sort.
There are different notions of infinite numbers. There are hyperreal numbers, ordinal numbers, cardinal numbers, one can view real numbers as infinite sequences of rationals and so infinitely more accurate than rational numbers (just as infinitesimals give us the ability to be more accurate than real numbers).
These notions grew out of some place where they were needed, and sometimes these places are somewhat orthogonal or unrelated (at least not directly).
In this case, we consider cardinals vs. hyperreal numbers. The cardinalities (under the axiom of choice, without the axiom of choice this is an even bigger mess, however with somewhat surprising results - more on that later) are well ordered. This means that between $\aleph_0$ and $\aleph_1$ there are no other cardinals. There are ordinals but they are all countable as sets.
Suppose you could somehow identify an element, call it $B$, in the hyperreals, $^*\mathbb R$, to be $\aleph_0$. What is $B+1$? In cardinalities $\aleph_0+1=\aleph_0$. In the hyperreals this is impossible.
What you could say is that all the elements of the form $B\pm n$ are still "infinite numbers".
The same problem would be met if we chose instead to identify ordinal numbers, since $1+\omega=\omega$ (where $\omega$ is the ordinal representing the natural numbers), this addition is non-commutative, as well not inversible, since subtraction is even less nice than addition when applied on ordinal numbers. 
A nice thing to consider: without the axiom of choice there can be cardinal numbers which are not $\aleph$-numbers (that is cannot be well ordered). It is consistent to have $2^{\aleph_0}$ cardinals so that they are ordered (by the natural ordering of cardinalities) like the real numbers. I would expect that it is possible to have that result extended to the hyperreal numbers. (Obviously we cannot expect cardinal addition and multiplication to be anything reversible)
This means that you can interpret the real numbers as cardinals. However none of them is even comparable with $\aleph_0$ (that is, none contain a countable subset).
(You might be interested in my answer here: "Homomorphism" from set of sequences to cardinals? which seems as though it may be somewhat relevant to this discussion.)
