  Gamma-ray bursts (GRBs) serve as powerful probes of the early Universe, with
their luminous afterglows revealing the locations and physical properties of
star forming galaxies at the highest redshifts, and potentially locating first
generation (Population III) stars. Since GRB afterglows have intrinsically very
simple spectra, they allow robust redshifts from low signal to noise
spectroscopy, or photometry. Here we present a photometric redshift of z~9.4
for the Swift-detected GRB 090429B based on deep observations with
Gemini-North, the Very Large Telescope, and the GRB Optical and Near-infrared
Detector. Assuming a Small Magellanic Cloud dust law (which has been found in a
majority of GRB sight-lines), the 90% likelihood range for the redshift is 9.06
< z < 9.52, although there is a low-probability tail to somewhat lower
redshifts. Adopting Milky Way or Large Magellanic Cloud dust laws leads to very
similar conclusions, while a Maiolino law does allow somewhat lower redshift
solutions, but in all cases the most likely redshift is found to be z>7. The
non-detection of the host galaxy to deep limits (Y_AB >~ 28 mag, which would
correspond roughly to 0.001 L* at z=1) in our late time optical and infrared
observations with the Hubble Space Telescope strongly supports the extreme
redshift origin of GRB 090429B, since we would expect to have detected any
low-z galaxy, even if it were highly dusty. Finally, the energetics of GRB
090429B are comparable to those of other GRBs, and suggest that its progenitor
is not greatly different to those of lower redshift bursts.
