The text bellow is just an explanation of the title, if you understood the title you don't have to read the text.
Games are defined by the rule "If L and R are two sets of games, then { L | R } is a game". If we also add the rule "Each element of L must be strictly less than each element of R", then we get the surreal numbers, $$. By adding this rule, we loose some games, e.g $$ and $*$ are games that aren't in $$.
There are gaps in $$ and the completion of $$ is called $^$(see link at bottom for clarification). When you construct $^$ you get back some of the games you lost when you constructed $$. $$ is an example of a game that is also in $^$($*$ however is not in $^$).
My question is this: Do you get any brand new elements when you construct $^$, elements that weren't games first?
https://en.wikipedia.org/wiki/Surreal_number#Gaps_and_continuity
In the question, it was written that "$\mathbf{On}$ and $*$ are games", but $\mathbf{On}$ is a gap that is not a game, because you can't get something equal to $\mathbf{On}$ with mere sets (as opposed to proper classes) in the left and right positions.
Any game with sets in the left and right positions is less than, say, the game/surreal $x=\{\kappa^+|\,\}$ where $\kappa$ is the cardinality for the depth of the game tree. And $x<\mathbf{On}$.
The gap $\infty$ mentioned in the wikipedia page is also not equal to a game, but that's less immediate.
(Also, readers should be careful not to confuse the gap $\mathbf{On}=\{\mathbf{No}|\,\}$ with the loopy game $\mathrm{on}=\{\mathrm{on}|\,\}$.)