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 Mumford-Tate 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.]