Largest Infinitesimal Game

Question

In combinatorial game theory, you can have infinitesimal games like up={0|*}={0|{0|0}}, and if you are allowed to use expressions that involve themselves as an option, like Off={|Off}, then you can find the smallest positive game: Tiny={0|{0|Off}} in the sense that there is no game that is less than Tiny but greater than 0.

What I want to know is: Is there a game which is in that sense the largest infinitesimal? In other words, does there exist a (infinitesimal) game G such that there is no other game H greater than G but less than all games with a real number value?

Note: There has been some confusion in the answers on what type of combinatorial games I am asking about. I want to know about the largest infinitesimal game out of all combinatorial games, including loopy and transfinite games.

Answer 1

There is no largest infinitesimal game, because if $G$ is a positive infinitesimal game, then $G+G$ is larger, but still infinitesimal.

However, in Chapter 16 of ONAG, in the section "The Gamut Revealed", we learn that the "largest infinitesimal games" are {$\alpha\, |\, \mathbb{R}^+ || \, \mathbb{R}^+\}$ where $\alpha$ is an ordinal. I believe this means that any infinitesimal game is smaller than some game of this form, for some choice of ordinal $\alpha$.

Answer 2

The original question brings up tiny, so the context is that of loopy games. If we restrict ourselves to finite loopy games, rather than the transfinite games Ted mentions in their answer, then the answer is "yes".

In that context, over={0|over} is the largest positive infinitesimal, being the infimum of the numbers of the form $1/2^n$ for integers $n$. Note: over + over = over, saving us from the "$G+G>G$" worry that Ted brought up.

over is also the supremum of numbers of the form $n\cdot\uparrow$, which answered the revised question "Is there a game $G$ such that for any infinitesimal $H$ there is an $n$ such that $H<nG$?" if you restricted things to the finite loopfree games.

Siegel's "Combinatorial Game Theory" is a great source for all of the above.