  Pink has given a qualitative answer to the Mumford-Tate conjecture for
Drinfeld modules in the 90s. He showed that the image of the v-adic Galois
representation is v-adically open in the motivic Galois group for any prime v.
In contrast to this result, we provide a family of uniformizable Anderson
t-modules for which the Galois representations of their t-adic Tate-modules are
"far from" having t-adically open image in their motivic Galois groups.
Nevertheless, the image is still Zariski-dense in the motivic Galois group
which is in accordance to the Mumford-Tate conjecture. For the proof, we
explicitly determine the motivic Galois group as well as the Galois
representation for these t-modules.
