Let $S$ be a compact orientable surface of genus $g \geq 2$. Is there any transversely real (or complex) analytic codimension one foliation $\mathcal{F}$ such that $\mathcal{F}$ has $S$ as a leaf with an infinite holonomy group?
Sure, take a homomorphism of $\rho:\pi_1(S)\to Diff^\omega(\mathbb{R})$ which has a global fixed point (for example, it might factor through $\mathbb{Z}$). Take the diagonal quotient $\mathbb{H}^2\times \mathbb{R}$, where $\pi_1(S)$ acts diagonally (on $\mathbb{H}^2$ as a fuchsian group, and on $\mathbb{R}$ by $\rho$). Then the quotient will be an analytic manifold with a closed surface leaf with nontrivial holonomy.

$\begingroup$ Is it possible to improve the example such that $\rho$ is injective? $\endgroup$ Sep 23 '14 at 20:14

$\begingroup$ I suspect so, but I don't know of an example immediately. $\endgroup$– Ian AgolSep 24 '14 at 3:01