Elementary proof that $\pi$ is transcendental A popular (and maybe the only) approach to showing that $\pi$ is transcendental is to first prove that for every non-zero algebraic number $a$, the number $e^a$ is transcendental. 
That requires tools from complex analysis.
But is there a known elementary proof that $\pi$ is transcendental? By an elementary proof I mean proof that does not use complex analysis. 
For example, there are known proofs that $e$ is transcendental which do not use complex analysis.
Also, can it be proved that complex analysis must be used to prove some given theorem?
 A: Not an answer, just a few comments that got too long.
The original (cleaned to modern standards) proof that $e$ is transcendental is just stuff with polynomials and the exponential on the reals, so not sure if it is accurate to claim that it uses "complex analysis"; even for $\pi$, the complex analysis used is minimal (maximum modulus essentially and maybe the stuff about rational conjugates being potentially complex numbers (like for $2^{1/3}$ say) but the question is fair; 
This being said this question on the need to use complex analysis or if you want the possibility of not using complex analysis, was popular in the 1920's regarding PNT and put to rest by Selberg/Erdos (though their proof of the PNT was much harder than the complex analysis one and that is still true today after major simplifications), but imho it misses the point as complex analysis usually simplifies things and purely real modern techniques are quite hard as one finds out easily by looking at harmonic functions in $\ge 3$ dimensions where the harmonic conjugate is not available anymore and proofs are sometimes much harder, so not sure what is the point
