There are three examples which I'm aware of.
GC Lie groups: these are Lie groups equipped with GC structures compatible with the multiplication map. Holomorphic Poisson Lie groups are an example of this, but there are others. For example, the known examples of generalized Kahler structures on compact even-dimensional semisimple Lie groups (we just need a bi-invariant metric, not all hypotheses are necessary) consist of two commuting GC structures, one of which is multiplicative in the above sense, and the other of which is a GC homogeneous space over the GC Lie groupd defined by the first. This situation will be familiar to those in Poisson Lie Group theory, and this is joint work in progress with Jiang-Hua Lu. David is of course correct in his statement that GC actions of GC Lie groups would then define GC action groupoids.
B-symplectic groupoids as described in http://arxiv.org/abs/math/0412097. These are, first and foremost, symplectic groupoids, but they have an extra 2-form making them GC Lie groupsgroupoids.
Any holomorphic Poisson groupoid is an example of a generalized complex groupoid. For example, if Z is a Poisson manifold, then $Z\times Z$ is a Poisson groupoid and hence a generalized complex groupoid.
