Are there Shimura Varieties of Hodgetype which are not of PELtype? I'd like to assume that the derived group is of type C.

$\begingroup$ Yes  PEL Shimura varieties are uncommon among those of Hodge type. Is the Shimura variety attached to $PGL_2$ of PELtype? $\endgroup$– abzJan 29 '14 at 4:37

$\begingroup$ @abz Thanks for pointing out my mistakes in my now deleted answer. I'd be glad to know a definitive answer to this question now. $\endgroup$– OlivierJan 31 '14 at 9:38

$\begingroup$ @abz thanks for your comment. What I was hoping to find out, though, was whether there exist Shimura Varieties of hodge type with no "extra endomorphisms" interpretation. From your answer, I'm guessing that there are plenty? $\endgroup$– user46141Jan 31 '14 at 22:02
Without the restriction on the lie algebra, certainly. There are interesting examples of families of abelian varieties with (generically) no extra endomorphisms but whose generic MumfordTate group is not the whole of $GSp_{2g}$.
I believe the first example, a family of fourfolds (the smallest dimension in which such a phenomenon arises) was given at the end of a short paper of Mumford "A note on Shimura's paper "Discontinuous groups and abelian varieties." http://dash.harvard.edu/bitstream/handle/1/3612771/McMullen_OnShimura.pdf?sequence=3
The idea is to take a division algebra $D$ over a cubic totally real field $F$ whose corestriction along $F/\mathbb{Q}$ is split, giving a natural map $Nm: D^* \rightarrow GL_8$. If $D$ ramifies at two of the three places at infinity, one shows that this map has symplectic image and can use the image to define a Hodge type Shimura datum. However, the representation is also absolutely irreducible which implies the underlying Hodge structure (hence abelian variety) attached to a general point in the family doesn't admit any extra endomorphisms.
However here I think the group is a $\mathbb{Q}$form of $SO(4) \times SL(2)$, which maybe isn't what you are looking for (though I suppose its Lie algebra is of type $C_1=A_1$).
[P.S. to abz: unless I'm mistaken, $PGL_2$ only gives a Shimura variety of abelian type.]