Does anyone know more examples of two nonisomorphic connected graphs with the same chromatic symmetric function? The only pair I know is the one in Stanley's paper on c.s.f.'s [http://math.mit.edu/~rstan/pubs/pubfiles/100.pdf; p.5 of the PDF file]. I would especially like to have an example in which at least one of the two graphs is trianglefree.

I don't think there are any other published examples. I think your best bet is to look at the literature on "chromatically equivalent graphs" (graphs with the same chromatic polynomial) and do your own computations to find examples. (I assume that you wrote some code to compute the chromatic symmetric function when you investigated trees.) 

