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We present the first integrated-light, TESS-based light curves for star clusters in the Milky Way, Small Magellanic Cloud, and Large Magellanic Cloud. We explore the information encoded in these light curves, with particular emphasis on variability. We describe our publicly available packageelk, which is designed to extract the light curves by applying principal component analysis to perform background light correction and incorporating corrections for TESS systematics, allowing us to detect variability on timescales shorter than ∼10 days. We perform a series of checks to ensure the quality of our light curves, removing observations where systematics are identified as dominant features, and deliver light curves for 348 previously cataloged open and globular clusters. Where TESS has observed a cluster in more than one observing sector, we provide separate light curves for each sector (for a total of 2204 light curves). We explore in detail the light curves of star clusters known to contain high-amplitude Cepheid and RR Lyrae variable stars, and we confirm that the variability of these known variables is still detectable when summed together with the light from thousands of other stars. We also demonstrate that even some low-amplitude stellar variability is preserved when integrating over a stellar population.