Model-completeness without quantifier elimination This is one of two mildly related questions
Are there examples of model-complete theories without quantifier elimination that do not reduce to theories expressed in the 'wrong' language? (E.g. Real closed fields without order relation or discrete linear orders without successor predecessor function.)
Note. A candidate could be Wilkie's theory. I do not master the details of that theory. In any case, I would like some basic example.
 A: Every theory has a definable expansion, called its Morleyization, to a theory with quantifier elimination. For every formula $\varphi(\overline{x})$, just add to the language a relation symbol $R_\varphi(\overline{x})$ and add to the theory an axiom $\forall\overline{x}\,R_\varphi(\overline{x})\leftrightarrow \varphi(\overline{x})$. 
So in this sense, every theory has quantifier elimination, once you express it in the "right" language. 
Of course, the issue with Morleyization is that it doesn't help you understand the formulas any better than you did before. In particular examples, a useful quantifier elimination result will tell us that a particular definable expansion of a theory $T$ has QE (or that every formula is equivalent to a Boolean combination of formulas of a certain simple form), and moreover it's possible to get some handle on what all the atomic formulas in the definable expansion (or what all of the formulas of that simple form) look like. 
So maybe what you really want is examples of model complete theories which have no "easy to understand" definable expansion with QE. This is a little vague, and I don't know the best place to draw the line. 
One option is just to ask for an example of a model complete theory $T$ such that any definable expansion of $T$ with QE must add infinitely many new symbols to the language. My understanding is that $\mathbb{R}_{\exp}$ is such an example, since every formula is equivalent to an existential formula of the form $\exists \overline{y}\,(\bigwedge_{i=1}^k f_i(\overline{x},\overline{y},e^{\overline{x}},e^{\overline{y}}) = 0)$, where each $f_i$ is a polynomial with integer coefficients, but there is no bound on the length of the quantified tuple $\overline{y}$.
