Single-resonant and (signal/idler) double-resonant optical parametric oscillators differ fundamentally on the conversion efficiency back to the pump wave. The nonpresent idler in the single-resonant case allows for signal intracavity enhancement well beyond the pump power level. This paper answers the question, how the phase-matched back conversion in a doubly-resonant system can be overcome to reveal substantial power enhancement, and what parameters it depends on. In a degenerate double-resonant OPO (DROPO) pumped by a thin-disk oscillator, an enhancement up to a factor of four is shown experimentally. Support of a semianalytical theory is presented with exceptionally simple relations between enhancement and intracavity losses. Interestingly, our theory predicts no fundamental limit to the maximal field enhancement or conversion efficiency.