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We tackle the problem of planning a minimum-time trajectory for a quadrotor over a sequence of specified waypoints in the presence of obstacles while exploiting the full quadrotor dynamics. This problem is crucial for autonomous search and rescue and drone racing scenarios but was, so far, unaddressed by the robotics community \emph{in its entirety} due to the challenges of minimizing time in the presence of the non-convex constraints posed by collision avoidance. Early works relied on simplified dynamics or polynomial trajectory representations that did not exploit the full actuator potential of a quadrotor and, thus, did not aim at minimizing time. We address this challenging problem by using a hierarchical, sampling-based method with an incrementally more complex quadrotor model. Our method first finds paths in different topologies to guide subsequent trajectory search for a kinodynamic point-mass model. Then, it uses an asymptotically-optimal, kinodynamic sampling-based method based on a full quadrotor model on top of the point-mass solution to find a feasible trajectory with a time-optimal objective. The proposed method is shown to outperform all related baselines in cluttered environments and is further validated in real-world flights at over 60km/h in one of the world's largest motion capture systems. We release the code open source.