What is the topology in $\Bbb R^3$ generated by planes in $\Bbb R^3$? 
What is the topology in $\Bbb R^3$ generated by planes in $\Bbb R^3$?

Let $\mathcal{A}$ be the collection of planes in $\Bbb R^3$. We have that $$\bigcup_{A \in \mathcal{A}} A = \Bbb R^3.$$ From this we can derive that the collection $\mathcal{A}$ is a subbasis  for $\Bbb R^3$. Now we can obtain a basis from the subbase $\mathcal{A}$ as follows $$\mathcal{B} = \bigcap\{A_k \mid k \in K \}$$ where $K$ is some indexing set. The intersection of a two planes in $\Bbb R^3$ is a line and the intersection of three planes is a point. Here it seems however, that I'm looking for the intersection of arbitarily many planes in $\Bbb R^3$? Is it so that these are either points or emptysets?
Following with the question if the basis is just a collection of singletons, then the generated topology would be the discrete one?
Perhaps there is another question arising here

Is the intersection of $n+m, m \in \Bbb N$ planes/hyperplanes in $\Bbb R^n$ a point in $\Bbb R^n$?

 A: In the topology generated by planes, every plane is an open set.
Once you have a topology in which every plane is an open set, you can conclude that every 1-point subset $\{x\} \subset \mathbb R^3$ is an open set, because there exist three planes whose intersection is $\{x\}$.
And once you know that every 1-point subset is open, then you can conclude that every subset is open. The topology is therefore discrete.

Regarding the basis $\mathcal B$ that is derived from the sub-basis $\mathcal A$, your formula is incorrect. The correct formula is
$$\mathcal B = \{A_1 \cap \cdots \cap A_k \mid A_1,\ldots,A_k \in \mathcal A\}
$$
In other words, you choose any finite indexed sequence of elements of the set $\mathcal A$, you intersect the elements of that sequence, and the result is an element of $\mathcal B$.
In the situation where $\mathcal A$ is the set of planes in $\mathbb R^3$, it follows from basic geometry that for any finite set of planes their intersection is either a plane, a line, a point, or empty. But all of that is not particularly relevant to the question of simply determining the topology generated by the set of planes; that is the discrete topology, as I showed.
