What's the deal with ace & pip? In Combinatorial Game Theory by Aaron Siegel on page 298 the games ace, deuce & trey are introduced:
$$\text{ace}=\{0|+_\text{on}\}$$
$$\text{deuce}=\{0|\text{ace}\}$$
$$\text{trey}=\{0|\text{deuce}\}$$
The book proceeds to assert that "in general":
$$\text{pip}_0=+_\text{on}$$
$$\text{pip}_{\text{n}+1}=\{0|\text{pip}_\text{n}\}$$
Additionally, the following relations are given:
$$\text{ace}+\text{ace}=\text{deuce}$$
$$\text{deuce}+\text{ace}=\text{trey}$$
$$n\cdot\text{ace}=\text{pip}_n (n\geq1)$$
After that, not much else is said about these games. My primary questions are:
A) What is pip?
B) How does ace compare to $+_\text{on}$? (Is it smaller, larger, equal or fuzzy? I would guess it's smaller, but I have read elsewhere that $+_\text{on}$ is the smallest of all games.)
C) What is the infimum/supremum of $\{\text{ace, deuce, trey,...}\}$? (My best guess is $\text{pip}_\text{on}$, however I'm unclear if this sequence is increasing or decreasing.) 
 A: What is pip?
The terminology here comes from words related to playing cards in a standard 52-card deck. Specifically, English nicknames, where "ace", "duece", and "trey" refer to the cards with one, two, and three symbols respectively. Those symbols are called "pips" (see this Oxford dictionary entry with the definition "Any of the spots on playing cards, dice, or dominoes."). The notation is suggestive of $\mathbf{pip}_n$ being a playing card with $n$ symbols on it. This comes from an extended playing card metaphor (introduced by the late John Horton Conway?) in the combinatorial game theory text Winning Ways for your Mathematical Plays (often abbreviated "Winning Ways").
How does $\mathbf{ace}$ compare to $+_{\mathbf{on}}$?
It's not hard to show that $\mathbf{ace}>0$. No matter if Left moves first or second, she wins by making the only available move. Then Proposition 2.12 on page 297 says we must have $\mathbf{ace}\ge +_{\mathbf{on}}$ ($+_{\mathbf{on}}$ is the smallest positive game).
Since $\mathbf{ace}\ne +_{\mathbf{on}}$, $\mathbf{ace}> +_{\mathbf{on}}$.
What is the infinimum/supremum of $\{\mathbf{ace},\mathbf{deuce},\mathbf{trey},\dots\}$?
Firstly, $\mathbf{pip}_G$ has not been defined for games $G$, so $\mathbf{pip}_{\mathbf{on}}$ doesn't really make sense. We only have $\mathbf{pip}_n$ for nonnegative integers $n$.
Since we have $\mathbf{ace}<\mathbf{deuce}<\cdots$, the minimum of your set is $\mathbf{ace}$, so that's the infimum.
The supremum is $\mathbf{over}=\{0\mid\mathbf{pass}\}$. You can guess this by looking at the trees for the $\mathbf{pip}_n$ in Figure 2.5. And one way to check it is from $\mathrm{aw}(\mathbf{pip}_n)=n$ on p299 and $\mathbf{over}=\sup_n(\uparrow n)$.
Calculating these things
As an aside, you can calculate with these pips using version 0.7 of Combinatorial Game Suite and the loopy game notation described here on cgsuite.org. For examples, you could use a command like tiny:={0|{0|off}};ace:={0|tiny};ace+ace>ace to confirm that $\mathbf{deuce}>\mathbf{ace}$, and it knows that a:{0|a} is $\mathbf{over}$.
