  The existence of a Kodaira fibration, i.e., of a fibration of a compact
complex surface $S$ onto a complex curve $B$ which is a differentiable but not
a holomorphic bundle, forces the geographical slope $ \nu(S) = c_1^2 (S) / c_2
(S)$ to lie in the interval $(2,3)$. But up to now all the known examples had
slope $ \nu(S) \leq 2 + 1/3$. In this paper we consider a special class of
surfaces admitting two such Kodaira fibrations, and we can construct many new
examples, showing in particular that there are such fibrations attaining the
slope $ \nu(S) = 2 + 2/3$. We are able to explicitly describe the moduli space
of such class of surfaces, and we show the existence of Kodaira fibrations
which yield rigid surfaces. We observe an interesting connection between the
problem of the slope of Kodaira fibrations and a 'packing' problem for
automorphisms of algebraic curves of genus $\geq 2$.
