"True" answer for MCMC model Theoretically, given a model with $N$ parameters and $\forall  x \in \mathbb{R}.\; p(x)>0$  in the prior of all parameters. If i'm interested only in the end result and not in time-to-convergence, given infinite amount on time, will I get the same posterior distribution for every possible choice of prior?
 A: No. There are two sorts of asymptotics here: $N$ items of data, and $T$ samples from the posterior. For most priors, as $N$ gets large the posterior distributions should become close*. For large $T$ you simply get a more accurate picture of the posterior under a particular prior for a fixed amount of data. 
*Asymptotic consistency of Bayesian models has some twists, I found a brief discussion in the intro here: http://www.stat.ufl.edu/archived/casella/Papers/AOS606.pdf . The case of whether the prior is parametric or nonparametric is important.
A: No it will not get the same posterior distribution for every possible choice of priors. Different priors lead to different posteriors (remember that the posterior $\alpha$ prior $\cdot$ likelihood) and thus 
chains from MCMC with different equilibrium distributions.
Convergence of MCMC is related to the ensurance that the built samples does not depend on the initial state (i.e. on the initialisation of the algorithm). So given a infinite amount of times, two runs of a same of MCMC (associated the same model, including the prior) both gives random samples from the same stationary law.
