Let $F$ be a closed oriented surface of negative Euler characteristic. Let $X^i(F)$ be the subset of the $SL\_2\mathbb{C}$-character variety of the fundamental group of $F$ corresponding to irreducible representations. The mapping class group of $F$, $\mathcal{M}(F)$ acts on $X^i(F)$. Let $[\rho]\in X^i(F)$, is the orbit of the $[\rho]$ under the action of the mapping class group,
$$\mathcal{M}(F).[\rho]$$ Zariski dense in $X(F)$?



Orbit of an irreducible representation of a surface group under that action of the mapping class group
Here is a partial answer (showing there exists a Zariski dense collection of irreducible representations with the requisite property).


In [*Topological Dynamics on Moduli Spaces II*](https://arxiv.org/abs/math/9910036) by Previte, and Xia it is shown in Theorem 1.4 that if the image of a $SU(2)$-representation of a surface group ($g\geq 2$) is dense in $SU(2)$ (which implies it's irreducible) then the mapping class group orbit of that representation in the $SU(2)$-character variety is dense (the result is stated in terms of relative character varieties but the closed case is included for genus at least 2 by fixing boundary values to be 2). 


Since the $SU(2)$-character variety is Zariski dense in the $SL(2,\mathbb{C})$-character variety, the property you want follows for the conjugacy classes of unitary representations with dense image.


Remark 1: In *[Representations of surface groups with finite mapping class group orbits](http://faculty.virginia.edu/Koberda/autpi.pdf)*, by Biswas, Koberda, Mj and Santharoubane, it is shown that for large $n$ there are $SL(n,\mathbb{C})$-representations with infinite image that are actually fixed by the entire mapping class group!


Remark 2: *[In Surface group representations in $SL\_2(\mathbb{C})$ with finite mapping class orbits](https://arxiv.org/abs/1707.00071)* by Biswas, Gupta, Mj, and Whang, it is shown (Theorem C) a representation has *bounded* mapping class group orbit in the character variety if and only if it is unitary up to conjugacy (if the genus $g\geq 2$).


