The further progress in thermoelectric applications strongly relies on development of new, more efficient thermoelectric materials. This has renewed the interest in transition-metal oxides (TMO), where strong electron correlations together with the degeneracy of electronic states can bring about a large thermoelectric power. Here we present a detailed study of thermoelectric as well as transport and magnetic properties of GdBaCo2O5+x, single crystals with layered perovskite-related crystal structure. A wide-range tunability of the oxygen content turns this compound into a unique filling-control system, which can be doped with either electrons (x < 0.5) or holes (x > 0.5), making this system attractive for elucidating the relation between the spin-orbital states and thermoelectric properties. We find a sharp divergence of the thermopower at x = 0.5, where it reaches a large absolute value (similar to 800 muV/K) and changes its sign from negative to positive upon increasing the oxygen content. The obtained experimental results indicate that in this strongly correlated transition-metal oxide the large thermoelectric power originates mainly from the entropy contribution of charge carriers, which is strongly enhanced due to the spin and orbital degeneracy.