Is it possible to obtain $\pi$ from finite amount of operations $\{+,-,\cdot,\div,\wedge\}$ on $\mathbb{N}$ (or $\mathbb{Q}$, the answer will still be the same)? Note that the set of all real numbers obtainable this way contains numbers that are not algebraic (for example $2^{2^{1/2}}$ is transcendental).
Bonus: If it happens that the answer is no, is it a solution to some equation generated that way (those $5$ operations performed finitely many times on elements on $\mathbb{N}$) ?