My favorite example is the 4-quantifier definition of an [almost-periodic function][1].  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.


  [1]: https://en.wikipedia.org/wiki/Almost_periodic_function