We investigate early black hole (BH) growth through the methodical search for z>~5 active galactic nuclei (AGN) in the Chandra} Deep Field South. We base our search on the Chandra 4-Ms data with flux limits of 9.1x10^-18^erg/s/cm2 (soft, 0.5-2keV) and 5.5x10^-17^erg/s/cm2 (hard, 2-8keV). At z~5, this corresponds to luminosities as low as ~10^42^erg/s/cm2 (~10^43^erg/s) in the soft (hard) band and should allow us to detect Compton-thin AGN with M_BH_>10^7^M_{sun}_ and Eddington ratios >0.1. Our field (0.03deg^2^) contains over 600 z~5 Lyman Break Galaxies. Based on lower redshift relations, we would expect ~20 of them to host AGN. After combining the Chandra data with Great Observatories Origins Deep Survey (GOODS)/Advanced Camera for Surveys (ACS), CANDELS/Wide Field Camera 3 and Spitzer/Infrared Array Camera data, the sample consists of 58 high-redshift candidates. We run a photometric redshift code, stack the GOODS/ACS data, apply colour criteria and the Lyman Break Technique and use the X-ray Hardness Ratio. We combine our tests and using additional data find that all sources are most likely at low redshift. We also find five X-ray sources without a counterpart in the optical or infrared which might be spurious detections. We conclude that our field does not contain any convincing z>~5 AGN. Explanations for this result include a low BH occupation fraction, a low AGN fraction, short, super-Eddington growth modes, BH growth through BH-BH mergers or in optically faint galaxies. By searching for z>~5 AGN, we are setting the foundation for constraining early BH growth and seed formation scenarios.