Counter-example of o-minimal structure but do not admit elimination of quantifiers I want a counter-example of o-minimal structure but do not admit elimination of quantifiers, we know that the inverse is true as (R,<,+,×,0,1) Tarski's theorem
 A: The classic example is $\mathbb{R}_{\text{exp}} = (\mathbb{R};<,+,-,\times,0,1,e^x)$. Wilkie showed that the complete theory of this structure is o-minimal and model complete, but does not admit quantifier elimination. Formulas of the form $\exists \overline{y}\,(f(\overline{x},\overline{y},e^{\overline{x}},e^{\overline{y}}) = 0)$, where $f$ is a polynomial with integer coefficients, are typically not equivalent to quantifier-free formulas. See Wilkie's Theorem on Wikipedia and the references linked there for more information.

This example, while natural, is not so elementary. Much more trivially, we can expand the real field by new relation symbols which are definable with parameters in $\mathbb{R}$, and hence do not break o-minimality, but which do break quantifier elimination.
For example, consider the structure $(\mathbb{R};<,+,-,\times,0,1,P)$, where $P$ is a binary relation which holds only of the single point $(\alpha,\beta)$, where $\alpha$ and $\beta$ are algebraically independent transcendental real numbers. Since $P$ is definable with parameters in the real field (by $x = \alpha\land y = \beta$), any formula with parameters in the expanded language is equivalent to a formula with parameters in the field language, so the expanded structure is o-minimal.
Now I claim that the formula $\exists y\,P(x,y)$, which defines the singleton $\{\alpha\}$, is not equivalent to any quantifier-free formula $\theta(x)$. Note that any atomic formula in a single variable $x$ which uses the relation symbol $P$ has the form $P(t(x),t'(x))$, where $t$ and $t'$ are terms in the language of fields (i.e., polynomials with integer coefficients). Since $\alpha$ and $\beta$ are algebraically independent and transcendental, there is no real number $\gamma$ and terms $t$ and $t'$ such that $t(\gamma) = \alpha$ and $t'(\gamma) = \beta$. Thus every atomic formula of the form $P(t(x),t'(x))$ defines the empty set in $\mathbb{R}$. So every quantifier-free formula in the single variable $x$ in the expanded language is equivalent to a quantifier-free formula in the language of ordered fields, by replacing instances of $P(t(x),t'(x))$ by $x\neq x$. Finally, note that since $\alpha$ is transcendental, the set $\{\alpha\}$ is not definable by a quantifier-free formula in the language of ordered fields (the definable elements in $\mathbb{R}$ are exactly the algebraic real numbers $\overline{\mathbb{Q}}\cap \mathbb{R}$).
