Let $t$ be transcendental over $K$, i.e. there is no $p(x) \in K[x]$ s.t. $p(t)=0$.
Can you define the smallest field containing $K$ and $t$ in a predicative way? If not, how would you prove that this is not possible?
The first definition coming to my mind is $K(t):= \bigcap_{K\subseteq L, t \in L } {L}$, with $L$ being a field. In this case, one of the fields $L$ must exactly be the object we are defining, thus making the definition impredicative.
Remark: the problem does not arise for the case of algebraic extensions. Given $a$ algebraic over $K$ with minimal polynomial $m_a(x) \in K[x]$, I thought about defining $K(a):= K[x]/(m_a(x))$, which is predicative. It's easy to show that this is a field containing $a$ and $K$, and a copy of $K(a):= K[x]/(m_a(x))$ can be embedded in any field containing $a$ and $K$, thus making $K[x]/(m_a(x))$ a good definition capturing "the smallest field containing $K$ and $a$".