Can cloud computing infrastructures provide HPC-competitive performance for scientific applications broadly? Despite prolific related literature, this question remains open. Answers are crucial for designing future systems and democratizing high-performance computing. We present a multi-level approach to investigate the performance gap between HPC and cloud computing, isolating different variables that contribute to this gap. Our experiments are divided into (i) hardware and system microbenchmarks and (ii) user application proxies. The results show that today's high-end cloud computing can deliver HPC-competitive performance not only for computationally intensive applications but also for memory- and communication-intensive applications - at least at modest scales - thanks to the high-speed memory systems and interconnects and dedicated batch scheduling now available on some cloud platforms.