True on-demand high-repetition-rate single-photon sources are highly sought after for quantum information processing applications. However, any coherently driven two-level quantum system suffers from a finite re-excitation probability under pulsed excitation, causing undesirable multi-photon emission. Here, we present a solid-state source of on-demand single photons yielding a raw second-order coherence of g(2)(0)=(7.5±1.6)×10-5 without any background subtraction or data processing. To this date, this is the lowest value of g(2)(0) reported for any single-photon source even compared to the previously reported best background subtracted values. We achieve this result on GaAs/AlGaAs quantum dots embedded in a low-Q planar cavity by employing (i) a two-photon excitation process and (ii) a filtering and detection setup featuring two superconducting single-photon detectors with ultralow dark-count rates of (0.0056±0.0007) s^-1 and (0.017±0.001) s^-1, respectively. Re-excitation processes are dramatically suppressed by (i), while (ii) removes false coincidences resulting in a negligibly low noise floor.