  We present imaging and spectroscopy of a hydrogen-poor superluminous
supernova (SLSN) discovered by the intermediate Palomar Transient Factory:
iPTF13ajg. At a redshift of z=0.7403, derived from narrow absorption lines,
iPTF13ajg peaked at an absolute magnitude M(u,AB)=-22.5, one of the most
luminous supernovae to date. The uBgRiz light curves, obtained with the P48,
P60, NOT, DCT, and Keck telescopes, and the nine-epoch spectral sequence
secured with the Keck and the VLT (covering 3 rest-frame months), are tied
together photometrically to provide an estimate of the flux evolution as a
function of time and wavelength. The observed bolometric peak luminosity of
iPTF13ajg is 3.2x10^44 erg/s, while the estimated total radiated energy is
1.3x10^51 erg. We detect narrow absorption lines of Mg I, Mg II, and Fe II,
associated with the cold interstellar medium in the host galaxy, at two
different epochs with X-shooter at the VLT. From Voigt-profile fitting, we
derive the column densities log N(Mg I)=11.94+-0.06, log N(Mg II)=14.7+-0.3,
and log N(Fe II)=14.25+-0.10. These column densities, as well as the Mg I and
Mg II equivalent widths of a sample of hydrogen-poor SLSNe taken from the
literature, are at the low end of those derived for gamma-ray bursts (GRBs),
whose progenitors are also thought to be massive stars. This suggests that the
environments of SLSNe and GRBs are different. From the nondetection of Fe II
fine-structure absorption lines, we derive a strict lower limit on the distance
between the supernova and the narrow-line absorbing gas of 50 pc. No
host-galaxy emission lines are detected, leading to an upper limit on the
unobscured star-formation rate of SFR([OII])<0.07 Msun/yr. Late-time imaging
shows the host galaxy of iPTF13ajg to be faint, with g(AB)~27.0 and R(AB)>=26.0
mag, which roughly corresponds to M(B,Vega) >~ -17.7 mag. [abridged]
