Is there any property about height of PDA?
If you could bound the stack height, say to some constant $c$, then it would have been possible to define an NFA for the task: Simply encode in the NFA states another $c$ values that represent the current values in the stack (by defining $\hat Q=Q\times\Sigma^c$. The rest i will leave to you)

This would contradict the non-regularity of this language, and therefore you cannot give such a bound.