More than the real numbers: hyperreals, superreals, surreals ...? I've read something about extensions of the real numbers, as hyperreals, superreals, surreals and, as I can understand, all these extensions contain some new kinds of infinitesimal and infinite ''numbers''.
And, if I well understand, we have a chain of inclusions as: $Reals \subset Hyperreals \subset Superreals \subset Surreals$ but, really, I don't well understand the difference between them.
So, before taking on a more detailed study, I have some ''naive'' question:
What kinds of elements are in one of these extensions but not in the other?
It is possible to have more ''great'' extensions or the chain stops with the Surreals?
These extensions have some utility in other branches of mathematic that justify the effort of study them?
 A: First of all, what you call "Reals, Hyperreals, Superreals and Surreals" are different type of objects. The real numbers form a unique field $\mathbb{R}$ (up to isomorphism), hyperreal fields and super-real fields form classes of non-isomorphic fields, and the surreal numbers form a unique Field $\mathbf{No}$ (up to isomorphism), meaning it is a proper class equipped with operations and an order that are also proper classes (informally, things too big to be sets).
Secondly, $\mathbb{R}$, $\mathbf{No}$ and all hyperreal fields and super-real fields are real-closed fields: fields with the same first order properties (first order sentences in the language of rings) as the field of algebraic real numbers. And all these fields can be ordered in a unique way. The class of real-closed fields is a very rich one, it has model-theoritic properties, geometric properties, set-theoritic properties, of course algebraïc properties, and an interesting - and still mysterious - spectrum of topological properties.
Thirdly, important properties of those fields are not given by the order type of their set infinitesimal elements, and one can't necessarly identify two ordered fields because they have isomorphic (let alone informally comparable) structures of infinitesimals. I would even say focusing on their types infinitesimals is a bad way to "understand" them.
I could tell you that there are infinitesimals such as $(1,\frac{1}{2},\frac{1}{3},...)$ in some hyperreal fields, as $\frac{1}{\omega_2}$ in the field of surreal numbers but I don't think it would do much more than entertaining you.

-The ordered field of real numbers is unique up to isomorphism as a field that satisfies the Least Upper Bound property. It is of course the field where the most part of analysis takes place, and the most part of continuous mathematics takes place in structures whose definitions involve $\mathbb{R}$ at some crucial point. It is also the field used by physicists.
The field $\mathbb{R}$ is archimedean, meaning every real number is less than some natural integer, and it is up to isomorphism the unique archimedean ordered field with no proper dense extension, the unique Cauchy-complete archimedean ordered field, the unique archimedean ordered field satisfying the Nested Interval Property. (I think you can find these facts and much more in Real Analysis in Reverse)
Quite importantly, every archimedean ordered field embeds in $\mathbb{R}$ in a unique way.
-Hyperreal fields are proper extensions of $\mathbb{R}$. As such, they are not archimedean, so there are elements in any hyperreal field greater than every positive integer or lower than every negative integers. Those elements are called infinite elements and their multiplicative inverses are called infinitesimals. They are $\aleph_1$-saturated, meaning that for any countable or finite subsets $A,B$ such that $A < B$, there is $x$ in the hyperreal field with $A < x$ and $x < B$. Hyperreal fields are interesting because of this property, and some of them can be used in Non-Standard Analysis or in model theory. In a certain sense, the field $\mathbf{No}$ is a hyperreal field.
You can find the definition of a hyperreal field on Wikipedia. I suggest you look for the simplest of them: ultrapowers of $\mathbb{R}$ (key-words: hyperreals, non-standard analysis).
-Super-real fields are also proper extensions of $\mathbb{R}$. They are more general than hyperreal fields, i.e. every hyperreal field is a super-real field. But they are not all $\aleph_1$-saturated. They were introduced mainly to enlarge the framework in which several open problems regarding Banach algebras could be anwswered.
-Finaly $\mathbf{No}$ is the only real-closed Field (up to isomorphism) such that for any subsets $A,B$ of $\mathbf{No}$ with $A < B$, there is an element $x$ of $No$ such that $A < x$ and $x < B$.
It is a beautiful inductive construction by John Horton Conway (other beautiful constructions exist, see Wikipedia) where there often is a way to define natural extensions of classical operations. It is also universal in the sense that every ordered field and every real closed Field embed in it. There are other Fields satisfying this.
$\mathbf{No}$ contains in a way "all kinds of infinitesimals"; but this only means that however you embed an ordered field $k$ in it, it will be contained in subfields of $No$ with infinitesimals a lot lower than that of $k$. However, there are proper non isomorphic subFields of $\mathbf{No}$ with infinitesimals as low as that of $No$.

If $k$ is an ordered field, you can always create a proper extension $k(X)$ of $k$ seen as the field of fractions with one indeterminate $X$ where the positive elements are fractions $F(X) = \frac{P(X)}{Q(X)}$ such that $\operatorname{sign}(P(X)) = \operatorname{sign}(Q(X))$. And $\operatorname{sign}(\sum \limits_{i=0}^d x_iX^i) = \operatorname{sign}(x_d)$.
If you do that with the field of surreal numbers $\mathbf{No}$, you get an extension $\mathbf{No}(X)$. The field $\mathbf{No}(X)$ is not real-closed, but it contains $\mathbf{No}$ and hence any ordered field (or Field). It also embeds into $\mathbf{No}$.

$\mathbb{R}$ is useful in mathematics in great part because (modern) mathematics use $\mathbb{R}$ as an elementary component to build more abstract objects. It is also (and this is historicaly important) the good field to model physical space (even though it is likely that an ultrafinitist point of view would be most accurate, $\mathbb{R}$ is at least a good homogenous approximation of space with every existence theorem we need). The are are also arithmetic statements about $\mathbb{Q}$ easier to prove in $\mathbb{R}$ once the general theorems have been established: the density of $\{x^n \ | \ x \in \mathbb{Q}_+\}$ in $\mathbb{Q}_+$ for instance.
I have read there are few theorems proven in IST (an instance of non-standard analysis) that are yet to be proven using regular analysis in ZFC.
I think $No$ is still at an early age of study (it isn't even forty years old), I don't know of any application of this study to other domains, even in field-theory (but I am not omniscient).  [edit] $\mathbf{No}$ is for now rather studied for itself (just like one might want to study complex numbers for themselves), and for its role as a "monster model" or sometimes just universal model of several well-behaved theories.
