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Aims. We present a detailed study of the superbubble 30 Dor C and the newly identified supernova remnant MCSNR J0536-6913 in the Large Magellanic Cloud. Methods. All available XMM-Newton data (flare-filtered exposure times of 420 ks EPIC-pn, 556 ks EPIC-MOS1, 614 ks EPICMOS2) were used to characterise the thermal X-ray emission in the region. An analysis of the non-thermal X-ray emission is also presented and discussed in the context of emission mechanisms previously suggested in the literature. These data are supplemented by X-ray data from Chandra, optical data from the Magellanic Cloud Emission Line Survey, and radio data from the Australia Telescope Compact Array and the Molonglo Observatory Synthesis Telescope. Results. The brightest thermal emission towards 30 Dor C was found to be associated with a new supernova remnant, MCSNR J0536-6913. X-ray spectral analysis of MCSNR J0536-6913 suggested an ejecta-dominated remnant with lines of O, Ne, Mg, and Si, and a total 0:3-10 keV X-ray luminosity of ~8*1034 erg s-1. Based on derived ejecta abundance ratios, we determined the mass of the stellar progenitor to be either ~18 M or as high as>40 M_sun, though the spectral fits were subject to assumptions (e.g., uniform temperature and well-mixed ejecta). The thermal emission from the SB exhibited enrichment by alpha-process elements, evidence for a recent core-collapse SNR interaction with the SB shell. We detected non-thermal X-ray emission throughout 30 Dor C, with the brightest regions being highly correlated with the H-alpha and radio shells. We created a non-thermal spectral energy distribution for the north-eastern shell of 30 Dor C which was best-fit with an exponentially cut-off synchrotron model. Conclusions. Thermal X-ray emission from 30 Dor C is very complex, consisting of a large scale superbubble emission at the eastern shell wall with the brightest emission due to MCSNR J0536??6913. The fact that the non-thermal spectral energy distribution of the superbubble shell was observed to roll-off is further evidence that the non-thermal X-ray emission from 30 Dor C is synchrotron in origin. [ABSTRACT FROM AUTHOR]