Proof that "$\uparrow$ is the unique solution of $\pmb{+}_G=G$" Tiny & miny games can be defined as
$$\pmb{+}_G = \{0||0|-G\}$$
$$\pmb{-}_G = -\pmb{+}_G = \{G|0||0\}$$
From the Wikipedia page for tiny and miny:

Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "$\uparrow$ is the unique solution of $\pmb{+}_G=G$"

$\uparrow$ is generally defined as $\{0|*\}$. I find it interesting that
$$\pmb{+}_0=\{0||0|0\}=\{0|*\}=\uparrow$$
$$\pmb{+}_0=\{0|0||0\}=\{*|0\}=\downarrow$$
How is $\uparrow$ the unique solution to $\pmb{+}_G=G$?
Additionally, if not apparent, what makes this amusing?
 A: I will use $\pmb{+}_G$ for $\text{tiny-}G$ throughout, but the unicode symbol ⧾ is arguably more correct for that plus sign.
Why Amusing?
This is a typo in the wikipedia page at the time of writing. On page 215 of On Numbers and Games, it actually says:

...it is amusing to verify that for any game $G$, we have $\pmb{+}_{\pmb{+}_{\pmb{+}_G}}=\uparrow$, so that in particular, $\uparrow$ is the unique solution of $G=\pmb{+}_G$.

Without asking, I can't be certain why John H. Conway found it amusing, but I personally find it amusing to work through why something complicated and with an arbitrary game parameter like $\pmb{+}_{\pmb{+}_{\pmb{+}_G}}\cong\{ 0\Vert0|-\{ 0\Vert0|-\{ 0\Vert0|-G\} \} \}$ simplifies to something as simple as $\uparrow\cong\{ 0\Vert0|0\}$.
Why $\uparrow$?
Verifying the claims in the quote above is a problem in Chapter 5 of Lessons in Play: An Introduction to Combinatorial Game Theory (tinies and minies are introduced in section 5.4). It's an amusing exercise, so I don't want to spoil the whole thing, but I can clarify the second part a bit.
Once we have $\pmb{+}_{\pmb{+}_{\pmb{+}_G}}=\uparrow$, then proving that $\uparrow$ is the unique solution (up to equality) to $\pmb{+}_G=G$ does not require any game theory. It's a general fact that if we have a function $f:X\to X$ and a particular $y\in X$ such that $\forall x\in X,f(f(f(x)))=y$, then $y$ is the only solution to $f(x)=x$. Can you see why?
