Dirac notation is ill-suited for non-self-adjoint operators. Here's why:
Let $(-,-)$ be the inner product on our Hilbert space. The expectation value of $AB$ is then
$$ \langle AB \rangle_\psi = (\psi,AB\psi)$$
by definition, and Dirac notation writes $\langle \psi \vert AB \vert \psi \rangle$. for this. But, in this notation, it is no longer clear to which side the operator $AB$ acts - one could as well interpret this expression as meaning $(BA\psi,\psi)$, which is not the same if $A,B$ are not self-adjoint. So, by
$$ \langle \psi \vert A \vert \psi \rangle^\ast = \langle \psi \vert A^\dagger \vert \psi \rangle$$
you really mean
$$ (\psi,A\psi)^\ast = (A\psi,\psi) = (\psi,A^\dagger\psi)$$
where the last equality is by definition of the adjoint.
So, examining the expression with $AB$, we find
$$ (\psi,AB\psi)^\ast = (AB\psi,\psi) = (\psi,(AB)^\dagger\psi) = (\psi,B^\dagger A^\dagger\psi)$$
$$ \langle \psi \vert AB \vert \psi \rangle^\ast = \langle\psi\vert B^\dagger A^\dagger \vert \psi \rangle $$
if all operators are interpreted as acting on the states to their right. However, since this is not usually understood - for self-adjoint operators it doesn't matter, and many texts freely switch the direction of the action of the operators whenever convenient - you should refrain from using Dirac notation for operators which are not self-adjoint.