This paper proposes a new quadratic version of a non-isolated extended-duty-ratio (EDR) boost converter for ultra-high gain applications. The conventional EDR topology is a combination of an interleaved inductor and switched capacitor configuration which attains high gain with significantly lower switch voltage stress. Quadratic boost converters are typically used for high gain applications with lower component count. In this paper, we have proposed a Quadratic EDR (Q-EDR) boost converter which is capable to attain ultra high gain while retaining the advantages of EDR boost. Two configurations of the proposed Q-EDR topology are presented to achieve equal current sharing among all switches with low input current ripple. A comprehensive converter operation analysis with design considerations and loss breakdown is presented. The performance of the proposed topology is validated using a hardware prototype operating at 24 V-36 V to 600 V at 100 kHz, for a 1 kW 4-phase Q-EDR boost with both single input inductor and interleaved inductor configurations. Experimental results are presented with the converter able to achieve a conversion ratio of 25 times at moderate levels of operational duty and device voltage stress limited to 150 V for 600 V output condition.