motivating examples of family of Hodge structure Let $\phi: \chi \to B$ be a proper holomophic submersion with smooth fiber $X$.
Then, we get local systems $R^i \phi_* \mathbb C$ and corresponding flat connection $(B, \bigtriangledown)$
In this case,there are tons of beautiful constructions even in such an elementary level: 
infinitesimal VHS, Mixed Hodge structure, Period mapping $P^{n,k} : B \to Grass(b^{n,k}, H^n(X, \mathbb C)), $ Picard-Lefschetz monosromy representation $\rho: \pi_1(B, b_0) \to GL(n, \mathbb C) $ and so on.
But my knowledge of these topic remains too abstract to digest it well. So i am collecting enlightening toy examples. For example, I've worked with the Legendre family of elliptic curves
{$y^2=x(x-1)(x-\lambda)$} $ \to $ {$\mathbb C -(0,1)$}
and interpreted everything into a concrete term.(and it was fantastic)
But i still wants more. Because in my examples, no mixed Hodge structure, no Hodge structure of weight $\ge$ 2. If you have any other good examples, please tell me. Good reference will be extremly helpful. I also appreciate any suggestion. 
 A: I guess the implied question is: what are good references containing  explicit calculations of variations of Hodge structure etc.? I might suggest taking a look at Griffiths' early pioneering papers "On periods of certain rational integrals I, II" Annals 1969, and
"Periods of integrals on algebraic manifolds III" IHES 1970. These papers contain a large number of explicit calculations on VHS and intermediate Jacobians for things like hypersurfaces in projective space. Regarding 
(variations of) mixed Hodge structures, take a look at the books by Carlson-Müller Stach-Peters, Peters-Steenbrink, and Voisin.
A: Sorry, I misread 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. 
