My favorite example is the 4-quantifier definition of an almost-periodic function. A first definition is that $f$ is almost-periodic iff $$\forall \epsilon\, \exists t>0\ \forall x\, |f(x+t)-f(x)| < \epsilon.$$
But $<$ for reals is a defined term, so if we use $x_n$ as the $1/n$ rational approximation of $x$, this means that $f$ is almost-periodic iff
$$\forall m\, \exists t>0\ \forall x\, \exists n\, \big|f(x+t)_n-f(x)_n\big| + \frac{2}{n} < \frac{1}{m}.$$
So Bohr's theorem that every almost-periodic function has arbitrarily good appropximations by trigonometric polynomials is a 5-quantifier statement.