Here is a comparison of lmer vs Stan output on a mildly complicated dataset from a psychology expt. (Kliegl et al 2011). The data are here: https://www.dropbox.com/s/pwuz1g7rtwy17p1/KWDYZ_test.rda.
The data and paper available from: http://openscience.uni-leipzig.de/index.php/mr2
I should say that datasets from psychology and psycholinguistic can be much more complicated than this. So this was only a modest test of Stan.
The basic result is that I was able to recover in Stan the parameter estimates (fixed effects) that were primarily of interest, compared to the lmer output. The sds of the variance components all come out pretty much the same in Stan vs lmer. The correlations estimated in Stan are much smaller than lmer, but this is normal: the bayesian models seem to be more conservative when it comes to estimating correlations between random effects.
Traceplots are here: https://www.dropbox.com/s/91xhk7ywpvh9q24/traceplotkliegl2011.pdf
They look generally fine to me.
One very important fact about lmer vs Stan is that lmer took 23 seconds to return an answer, but Stan took 18,814 seconds (about 5 hours), running 500 iterations and 2 chains.
One caveat is that I do have to try to figure out how to speed up Stan so that we get the best performance out of it that is possible.