Zero (according to my personal scale, and subject to the caveat that the universe is not at equilibrium).
The Gibbs free energy $G=U-TS+PV$. For a well-defined system in equilibrium, the temperature $T$, entropy $S$, pressure $P$, and volume $V$ all have single absolute values; the internal energy $U$ has a single value, but it is not absolute; it can only be measured relative to an arbitrary reference state. So I'm free to set the Gibbs free energy of the universe equal to zero and work from there, although I may run into problems with my assumption of equilibrium.
(The enthalpy $H=U+PV$ also can only be defined relative to an arbitrary reference state, and the same is true for the Helmholtz free energy $A=U-TS$ and the chemical potential $\mu=(dG/dN)_{T,P}$, for example.)