Suppose $P_2(n)$ is the set of all $n$-ary boolean functions, $P_2 := \bigcup_{n \in \mathbb{N}} P_2(n)$. Now for $A \subset P_2$ define $[A]$ as the minimal boolean clone containing $A$ as a subset. Let's define $Cl_2(n) := \{[A]|A \subset P_2(n)\}$
Is there a way to explicitly calculate $|Cl_2(n)|$? Or (if it is too complicated) what is its asymptotic?
The only thing I was able to get was the following upper bound:
$$|Cl_2(n)| \leq 2^{2^{n^2}}$$
It follows from $|P_2(n)| = 2^{n^2}$.
First of all, it is not true that $|P_2(n)|=2^{n^2}$, rather $|P_2(n)|=2^{2^n}$. This affects your estimate for $|Cl_2(n)|$.
Second, it is possible to compute $|Cl_2(n)|$ exactly using Post's lattice. I won't attempt to calculate $|Cl_2(n)|$ for $n=0,1,2,3$, but for $n\geq 3$ it is easy to see from Post's lattice that $|Cl_2(n+1)|=|Cl_2(n)|+8$. Thus $|Cl_2(n)|=8n+C$ for some constant $C$ whenever $n\geq 3$.