Less than all positive numbers, greater than all negative numbers, and not zero; what is $\ast$? A part two, you could say, of my previous question.
I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of ViHart said the following -

Star. Which, it's not a number. It's like a number. The thing is, it's
  smaller than all the positive numbers, and it's larger than all the
  negative numbers. But it's not zero. It gets confused with zero.
This is one of those weird things you get in combinatorial game
  theory, and, I like it. I think the very non-numberness of it, I think
  that's what appeals about a lot of these numbers.

The video itself can be found here.
I've searched the internet for anything about star, or $\ast$, as she wrote, but I can't seem to find anything that explains it well. All I understand is that it has something to do with combinatorics.
Would anyone know what it is, it's application, it's properties, or what it means, in terms that are possibly more clear than those found on Wikipedia, or any other sources that pop up from searching for star?
 A: Don't think of $\ast$ as a number to begin with; that will only confuse you. $\ast$ is a mathematical object called a combinatorial game. An example of a combinatorial game you may have played before is Nim. In general, combinatorial games are characterized by having two players who take turns making moves, and also by the fact that both players have perfect information about the game (so a card game where you don't know your opponent's hand is not a combinatorial game). 
$\ast$ itself is a pretty boring game: the player who moves first wins. 
So:


*

*Why does anyone care about $\ast$? In combinatorial game theory, combinatorial games admit a very elegant recursive definition: they are defined in terms of other games! Using the methods of combinatorial game theory, you can analyze complicated games in terms of simpler games, and sometimes when doing this $\ast$ appears as one of the simpler games you analyze. 

*Why did Vi Hart say that $\ast$ is "like a number"? It turns out that combinatorial games behave like numbers in many ways: you can add them as well as say when one game is greater than or equal to another game. There are also some combinatorial games that behave like the usual numbers you're used to, but others that behave very differently, like $\ast$. 
To really understand this situation I recommend that you learn a little abstract algebra and number theory, which will teach you to think about various kinds of numbers. As far as combinatorial game theory itself, my understanding is that the classical texts here are Conway's On Numbers and Games and Berlekamp, Conway, and Guy's Winning Ways for your Mathematical Plays. 
If you play chess or go, combinatorial game theory can be applied to understanding chess and go endgames (see Elkies for the former and these resources for the latter). In particular, you can use chess or go positions to write down some of the weirder combinatorial games that don't behave like numbers. 
A: The version of $*$ I know is described in  Winning Ways for your Mathematical Plays by  Elwyn R. Berlekamp, John H. Conway and Richard K. Guy.  It is a terrific book.  They start by describing games as sets that look like $\{|\}$.  The things to the left of the bar are options of one player, called Left.  The things to the right of the bar are options for Right.  The empty set I showed above is naturally zero, as whoever has to move loses-he has no options.  $+1$ is represented as $\{0|\}$, so Left has a move but Right does not.  Any number greater than zero is a win for Left, whoever starts.  Any number less than zero is a win for Right, whoever starts.  $*=\{0|0\}$, so either player can move to zero and win.  But $*+*=0$ because in the sum of two stars, the first player will move in one, the other will move in the other and the first player loses, which was our definition of zero.
