The characteristic physical timescales near stellar-mass compact objects are measured in milliseconds. These timescales -- the free-fall time, the fastest stable orbital period, and stellar spin periods -- encode the fundamental physical properties of compact objects: mass, radius, and angular momentum. The characteristic temperature of matter in the vicinity of neutron stars is such that the principal electromagnetic window into their realms is the X-ray band. Because of these connections to the fundamental properties of neutron stars, X-ray timing studies remain today the most direct means of probing their structure and dynamics. While current X-ray observatories have revealed many relevant and fascinating phenomena, they lack the sensitivity to fully exploit them to uncover the fundamental properties of compact objects and their extreme physics. With this white paper, we summarize and highlight the science opportunities that will accompany an order-of-magnitude improvement in X-ray timing sensitivity, a goal attainable in the coming decade.
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