It looks like the following very important example is not still mentioned here.
Pigeonhole principle plays a crucial role in K. F. Roth's proof that for any $\kappa>2$ and any algebraic irrational real number $\alpha$ inequality $|\alpha-p/q| < q^{-\kappa}$ has only finitely many solutions for rational fractions $p/q$.
(Lemma 9 in: K. F. Roth, Rational approximation to algebraic numbers, Mathematika 2 (1955), 1-20).
