The function $SSCG(k)$ does not give a set of graphs, it is a function that takes in a natural number $k$, and returns a natural number $SSCG(k)$ (we now explain how).
A graph is said to be a simple subcubic graph if it is a simple graph in which every vertex has degree at most $3$. To explain $SSCG$, we introduce our own new definition.
Let $S = (G_1, G_2, \dots, G_n)$ be a sequence of graphs. The length of the sequence is the number of graphs in it. We call the sequence $k$-valid if it satisfies the following things:
- Every graph $G_i$ is a simple subcubic graph,
- The graph $G_i$ has at most $i+k$ vertices $(|V(G_i)|\leq i+k)$,
- Each graph $G_i$ does not have any graph that comes before it as a minor (So if $i<j$, then $G_i$ is not a minor of $G_j$).
By a version of the Robertson-Seymour Theorem, there is a finite upper bound on the length of a $k$-valid sequence. So we define $SSCG(k)$ to be the maximum length among all $k$-valid sequences. Note that, for whatever reason, the empty graph $\emptyset$ with no vertices is counted as a graph.
To understand what's going on here, let's calculate $SSCG(0)$ and $SSCG(1)$ explicitly.
To find $SSCG(0)$, it suffices to find all $0$-valid sequences.
The first graph $G_1$ of a $0$-valid sequence has at most $1+0 = 1$ vertices. Since $G_1$ is simple, it must either be the empty graph, or $K_1$. The empty graph is a minor of every graph, so no graph can come after it in a valid sequence. So the sequence with $G_1=\emptyset$ has length $1$.
But what if $G_1=K_1$? Note that $K_1$ is a minor of every graph that has at least one vertex, so the only graph that can come after $K_1$ in a valid sequence is the empty graph, and we get the sequence $(G_1, G_2) = (K_1, \emptyset)$ with length $2$. This exhausts all possibilities, so we see that the max length of a $0$-valid sequence is $2$, so $SSCG(0) = 2$.
Now let's try $SSCG(1)$. By requirement 2 for a $1$-valid sequence, we know that the first graph $G_1$ of any $1$-valid sequence has at most $1+1 = 2$ vertices. So our only possibilities are $\emptyset, K_1, K_2$ and $I_2$ (where we let $I_k$ be the graph with $k$ vertices and no edges). By the previous argument, we know any sequence starting with $\emptyset$ or $K_1$ has length at most $2$.
So let's consider a sequence starting with $G_1 = I_2$. Since $I_2$ is a minor of any graph with $2$ or more vertices, we must have $|V(G_2)| < 2$, so $G_2 = K_1$ or $G_2 = \emptyset$ and we see that any sequence starting with $I_2$, such as $(I_2, K_1, \emptyset)$ has length at most $3$.
Finally, consider the $1$-valid sequences that start with $G_1 = K_2$. The graph $K_2$ is a minor of any graph that has at least one edge, so $G_2$ cannot have any edges (nor can $G_i$ for any other $i>1$). We know that $G_2$ must have at most $2+1 = 3$ vertices, so $G_2 \in \{K_1, I_2, I_3\}$. We have already shown that if our sequence has $I_2$, we can only get another $2$ graphs in it ($K_1$ and $\emptyset$), so we try $G_2 = I_3$. Because $I_3$ is a minor of every graph with at least $3$ vertices, we have that $G_3$ has at most $2$ vertices and no edges. The best we can do is to set $G_3 = I_2$. Using the previous arguments, this gives us a sequence $(K_2, I_3, I_2, K_1, \emptyset)$ of length $5$, and we can't do better. So $SSCG(1) = 5$.
I found this page a lot more useful than the wikipedia page for understanding this.
Best Answer
The busy beaver function $BB(n)$ is (informally) an upper bound on the amount of time a computer of size $n$ (that is, an $n$-state Turing machine) can compute without going into an infinite loop. It increases much more quickly than does Ackermann's function; so much so that it can't be computed at all. In fact, it increases more quickly than any function that can be computed.
The busy beaver function shows up in all sorts of examples of non-computability. For example, Are there any examples of non-computable real numbers? asked for an example of a real number that can't be computed by any process, and among the answers was $$\sum_{i=1}^\infty 2^{-BB(i)}=2^{-1}+2^{-6}+2^{-21}+2^{-107}+\ ... \ \approx 0.515625476837158203125000000000006$$ in which the busy beaver function plays a central role.