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.

My favorite example is the 5-quantifier definition of an almost-periodic function: $f$ is almost-periodic iff $$\forall \epsilon>0\, \exists t>0\ \forall a\ \exists s \in [a,a+t]\ \forall x\, |f(x+s)-f(x)| \leqslant \epsilon.$$

Using the fact that almost periodic functions are uniformly continuous, one can show that this is equivalent to a 3-quantifier property. The 5-quantifier definition remains the more natural one, especially for the purpose of stating the above fact.

So Bohr's theorem that every almost-periodic function has arbitrarily good appropximations by trigonometric polynomials is naturally written as a 6-quantifier statement.

PS Thanks to @FrancoisZiegler and @DmytroTaranovsky for sorting me out on this.