experimentally two-qubit entangling gates at nearly the fastest possible speed allowed by the physical interaction strength between two superconducting transmon qubits. We achieve this quantum speed limit by implementing experimental gates designed using a machine learning inspired optimal control method. Importantly, our method only requires the single-qubit drive strength to be moderately larger than the interaction strength to achieve an arbitrary entangling gate close to its analytical speed limit with high fidelity. Thus, the method is applicable to a variety of platforms including those with comparable single-qubit and two-qubit gate speeds, or those with always-on interactions.