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We obtained an LBT/PEPSI spectrum with very high resolution and high signal-to-noise ratio (S/N) of the K0V host Kepler-444, which is known to host 5 sub-Earth size rocky planets. The spectrum has a resolution of R=250,000, a continuous wavelength coverage from 4230 to 9120A, and S/N between 150 and 550:1 (blue to red). We performed a detailed chemical analysis to determine the photospheric abundances of 18 chemical elements, in order to use the abundances to place constraints on the bulk composition of the five rocky planets. Our spectral analysis employs the equivalent width method for most of our spectral lines, but we used spectral synthesis to fit a small number of lines that require special care. In both cases, we derived our abundances using the MOOG spectral analysis package and Kurucz model atmospheres. We find no correlation between elemental abundance and condensation temperature among the refractory elements. In addition, using our spectroscopic stellar parameters and isochrone fitting, we find an age of 10+/-1.5 Gyr, which is consistent with the asteroseismic age of 11+/-1 Gyr. Finally, from the photospheric abundances of Mg, Si, and Fe, we estimate that the typical Fe-core mass fraction for the rocky planets in the Kepler-444 system is approximately 24 per cent. If our estimate of the Fe-core mass fraction is confirmed by more detailed modeling of the disk chemistry and simulations of planet formation and evolution in the Kepler-444 system, then this would suggest that rocky planets in more metal-poor and alpha-enhanced systems may tend to be less dense than their counterparts of comparable size in more metal-rich systems., Comment: in press, 11 pages, 3 figures, data available from pepsi.aip.de