Short answer: This is not known. With the operations you allow (starting from N or Q and using only finitely many applications of +, −, ·, ÷, and exponentiation), it is an open problem whether one can obtain π, and it is also open whether π is a root of any nontrivial equation built from these same operations.

More context

• The set of numbers you can produce this way is countable (there are only countably many finite expressions over a countable alphabet). That by itself doesn’t tell you whether a particular number like π is in the set.

• Your class already contains many transcendental numbers, e.g. 2^{√2} by the Gelfond–Schneider theorem, and, if one allows complex intermediate values, e^{π} = (−1)^{−i} is also in the class and is transcendental. So “being transcendental” does not prevent membership.

• What you are asking in the “bonus” is whether π is a solution of a nonzero exponential-polynomial equation with rational coefficients, where “exponential-polynomial” here means a finite expression in the variable using only +, −, ·, ÷, and the binary operation (^) (allowing the variable to appear both in bases and exponents). This, too, is open.

• There are strong conjectures in transcendence theory (notably Schanuel’s conjecture) that would imply a negative answer to both questions: π could neither be generated by finitely many applications of +, −, ·, ÷, ^ starting from rationals, nor be a root of a nontrivial exponential-polynomial with rational coefficients. Very roughly, rewriting an expression involving a^b using exp and log introduces logarithms of algebraic numbers; Schanuel’s conjecture predicts very stringent algebraic-independence properties among such quantities, which would rule out accidental identities forcing π to be obtainable in your restricted sense or to satisfy a nontrivial exponential-polynomial equation over Q. However, these implications remain conditional on conjectures.

• Unconditionally, we do not currently have methods to prove that π cannot be produced in your way or that it cannot satisfy such an exponential equation. Known theorems such as Lindemann–Weierstrass and Baker’s theory give many transcendence and linear-independence results for values of exp/log at algebraic points, but they do not settle your question for π with only +, −, ·, ÷, ^ allowed.

So, as of now:

• Is π obtainable by finitely many uses of +, −, ·, ÷, ^ from N (or Q)? Unknown.
• Bonus: Is π a root of some nontrivial equation built from those operations with rational constants? Also unknown.
• Under Schanuel’s conjecture (and related conjectures), both answers would be no.

References for background:

• A. Baker, Transcendental Number Theory (for Gelfond–Schneider, linear forms in logarithms).
• M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups (surveying transcendence and conjectures such as Schanuel’s).
• A. J. Macintyre and A. J. Wilkie, On the decidability of the real exponential field (shows how Schanuel-type statements impact equations with exponentials).