Sorry, I am in a similar situationmisread the question: I thought you wanted examples with no mixed Hodge structure and no Hodge structure of weight $\geq 2$. I tried to delete this post, but it did not work.
I am definitely not an algebraic geometer, but I was recently forced to deal with some of the structures you mentioned in some very simple settings.
I have learned a lot from the paper "Braid Groups and Hodge Structures" by Curt McMullen: http://www.math.harvard.edu/~ctm/papers/home/text/papers/bn/bn.pdf. In my opinion, McMullen's papers (on any subject) are absolutely fantastic, and are a great pleasure to read.
In the same direction, some nice examples come from Teichmuller curves. You can check out e.g. the two preprints by Alex Wright: http://arxiv.org/abs/1203.2683 ("Schwarz triangle mappings and Teichmüller curves I: abelian square-tiled surfaces") and http://arxiv.org/abs/1203.2685 ("Schwarz triangle mappings and Teichmüller curves II: the Veech-Ward-Bouw-Möller curves").
Also, some shameless self-advertising: there is http://arxiv.org/abs/1112.5872
by M. Kontsevich, A. Zorich and me (about square-tiled surfaces) which is mostly expository.
