Say we have the language $L = \{C, P\}$ where $C$ is a nullary predicate and $P$ a unary one. Then we consider the empty theory $E$ over the language $L$. My task was to give an example of a Henkin conservative extension of E. However when I submitted:
$$T = \{\exists x P(x) \Rightarrow P(c), \exists x \neg P(x) \Rightarrow \neg P(d)\}$$
it was wrong. Why is that?
My understanding of the term "Henkin theory" is that $T$ is Henkin if for every formula$^1$ $\varphi$, $T$ proves $$(\exists x\varphi(x))\implies \varphi(c_{\varphi(x)})$$ for some (not necessarily unique) constant symbol $c_{\varphi(x)}$.
The point is that you haven't considered all possible formulas here. For any finite set of constant symbols $u_1,...,u_n$ that you've added, you also need to consider the formula saying that there is something not equal to any of them, and this will generate a new constant symbol $v$: $$\exists x(x\not=u_1\wedge...\wedge x\not=u_n)\implies v\not=u_1\wedge ...\wedge v\not=u_n.$$ So in fact we get "infinite blow-up" - even in the completely trivial case of the empty theory over the empty language (since we need to add a single constant for $\exists x(x=x)$).
$^1$Note that it is enough to consider unary formulas: to handle $\exists x\exists y\varphi(x,y)$, we get $$T\vdash (\exists x\exists y\varphi(x,y))\implies \varphi({c_{\exists y\varphi(x,y)}}, c_{\varphi(c,y)}).$$