On page 297 of Combinatorial Game Theory by Aaron Siegel it is stated that:
$+_{on}=\{0||0|\text{off}\}$ [is] the smallest positive game of all.
In More Infinite Games by John H. Conway the following is stated:
Numbers like $\omega^{1/\omega^a}$ are the smallest infinite numbers but not the smallest infinite games.
Conway also states:
Note that $\uparrow$ is not a number: it is the value of a game, which is a more subtle concept. Also note that $\frac{1}{\uparrow}$ is not defined since it would be bigger than all surreal numbers and there are no such numbers. (In fact, it does exist but is one of the Oneiric numbers.)
It also says:
$G=\{G|\}$ or $G=\{\text{pass}|\}$ and then by transfinite induction $G$ is bigger than zero, all ordinals and, in fact, any game, The game is called $On=\{On|\}$. [...] This game also gives $\frac{1}{On} = \{0|\frac{1}{On}\}$ [..] Therefore $\frac{1}{On}$ is absolutely the smallest positive game.
As far as I know, $\frac{1}{On} \neq +_{on}$. If they are equal, then it seems $On=\frac{1}{+_{on}}$. If $\frac{1}{On} < +_{on}$ then $On>\frac{1}{+_{on}}$. If $\frac{1}{On} > +_{on}$ then $On<\frac{1}{+_{on}}$.
So, what is the smallest game of all? What is the largest?
Additionally, does a hierarchy similar to the following hold for $n \geq 1$: $+_n \ll \uparrow^n \ll \frac{1}{\omega^n} \ll \frac{1}{\infty^n} \ll n \ll \infty^n \ll \omega^n \ll \frac{1}{\uparrow^n} \ll \frac{1}{+_n}$?
Disclaimers: It is not common to talk about loopy games and transfinite games at the same time, but a lot of what Siegel writes about finite loopy games still holds, mutatis mutandis, if we allow nonloopy games to be transfinite, thanks to the axiom of foundation or the equivalent.
Also, More Infinite Names is a very informal note that requires knowledge of a bunch of assorted references and the nonstandard and somewhat inconsistent notation Conway was using.
If we restrict loopy games to be finite , then the game $H$ satisfying $H=\left\{ H\mid\,\right\}$ (which Conway there calls $\mathrm{On}$ and Siegel calls $\mathbf{on}$) is certainly worth calling the largest game. Siegel's argument on p284 that $\mathbf{on}\ge G$ for all $G$ still works.
Given that the game Conway called $\dfrac{1}{\mathrm{On}}$ satisfies $H=\left\{ 0\mid H\right\}$ , it is the game that Siegel calls $\mathbf{over}$. I believe Conway's intent for writing it as $\dfrac{1}{\mathrm{On}}$ was simply to suggest it was the game infimum of $\left\{ \dfrac{1}{x}\mid x\text{ is a (surreal) number}\right\}$ . Division by a game that isn't a number, especially a loopy one like $\mathbf{on}$, is not standardly defined.
We do have $\mathbf{over}>+_{\mathbf{on}}$, but neither game is a number and so we cannot reciprocate either side.
On page 297 of Siegel, he confirms "$+_{\mathbf{on}}$ is the smallest positive game of all" in the finite context, and the argument works essentially the same way in our more general context.
For your question about $+_{n}\ll\uparrow^{n}\ll\frac{1}{\omega^{n}}\ll\frac{1}{\infty^{n}}\ll n\ll\infty^{n}\ll\omega^{n}\ll\frac{1}{\uparrow^{n}}\ll\frac{1}{+_{n}}$: As several games here are not numbers, we cannot form their reciprocal. $\frac{1}{\infty^{n}}$, $\frac{1}{\uparrow^{n}}$, and $\frac{1}{+_{n}}$ are not standardly defined. So the only thing we could possibly ask is: $+_{n}\ll\uparrow^{n}\ll\frac{1}{\omega^{n}}\ll n\ll\infty^{n}\ll\omega^{n}$. And indeed that is true. If there is a part of it whose proofs from CGT or elsewhere you need help with, I would be happy to clarify.
A side note: I would recommend you not bounce around between various sources when learning this stuff, as notations differ, and there might be corrections in newer sources. If you are not finding the answers to your questions from Siegel, then I would recommend going through an undergraduate text and then coming back to Siegel.