I apologise if this is a duplicate in any way. I couldn't find anything here.
In this Numberphile video at about six and a half minutes in, Vi Hart chooses the "number" $*$ as her favourite number. She says it's less than all the positive numbers and greater than all the negative numbers, but is not zero.
What is a formal definition of this "number" (if the above doesn't qualify as one)?
She says it's used in Combinatorial Game Theory. What is it used for exactly?
A combinatorial game is a game where two players (called Left and Right) take turns making moves and the first player who does not have a legal move loses (and the game is guaranteed to end after a finite but possibly unbounded number of moves). Such a game is represented with the notation $\{S\mid T\}$, where $S$ is the list of positions that Left can move to and $T$ is the list of positions Right can move to.
The symbol $*$ refers to the game $\{\{\mid\}|\{\mid\}\}$. To unravel this notation, let us first define $0=\{|\}$, the game in which neither player has a legal move (so whoever moves first loses!). Then $*$ is $\{0|0\}$: each player can move to $0$ on their turn. So $*$ is a game in which the player who moves first wins, since they move to $0$ and then the other player has no legal moves.
It turns out that games have a natural algebraic structure. First, there is an operation of game addition: given two games $x$ and $y$, their sum $x+y$ consists of playing $x$ and $y$ simultaneously, where on your turn you make a move in either one of the games. So for instance, in the game $*+*$, the first player picks one of the two copies of $*$ to move to $0$ in, giving the position $*+0$ or $0+*$. Then the second player must move in the other copy of $*$ (since they have no legal moves in $0$!) giving the position $0+0$. The first player now has no legal moves, and so the second player wins.
We can now define an equivalence relation on games: say two games $x$ and $y$ are equivalent if the second player to move always wins the sum $x+(-y)$, where $-y$ is the game obtained by swapping the roles of Left and Right in the game $y$. It is not obvious at first glance, but it turns out that this equivalence relation is very natural and powerful. If you take games modulo this equivalence relation, they form an abelian group under sum with $0$ as the identity element. (From now on, whenever I say "game", I will really mean "equivalence class of games" under this equivalence relation.)
Now games also have a natural partial order. Roughly speaking, we say $x>y$ if Left has a greater advantage in game $x$ than they do in game $y$. More precisely, we say:
In particular, note that $*$ is incomparable to $0$, since $*+(-0)=*$ and we saw the first player to move wins in $*$.
Now, what does this have to do with numbers? It turns out that certain games can naturally be identified as "numbers", with addition and the ordering on games corresponding to the usual addition and ordering on numbers. These numbers are known as the surreal numbers, and if you restrict to games with only finitely many positions, they can be naturally identified with the dyadic rational numbers. Roughly speaking, a game which is identified with a number $n$ is a game in which Left has an "$n$-move advantage" over Right. When restricted to numbers, our partial order $\leq$ on games is a total order. (There is also a "multiplication" operation that makes the numbers an ordered field, which is not defined for more general games.)
Now, the game $*$ is not a number. This follows from the fact that $*$ is incomparable to $0$ in our ordering (and so it cannot be a surreal number since the surreal numbers are totally ordered and contain $0$). However, it turns out that $*$ is "infinitesimal" in a strong sense. In particular, for any positive surreal number $x$, $*<x$, and for any negative surreal number $x$, $*>x$. (Note that the surreal numbers themselves have "infinitesimal" elements that are positive but smaller than all positive rational numbers, so $*$ is infinitesimal in an even stronger sense.)