We present the results of a multiwavelength survey of HI-excess galaxies, an intriguing population with large HI reservoirs associated with little current star formation. These galaxies have stellar masses M* > 1010 M⊙, and were identified as outliers in the gas fraction versus NUV-r colour and stellar mass surface density scaling relations based on the GALEX Arecibo SDSS Survey (GASS). We obtained HI interferometry with the Giant Metrewave Radio Telescope, Keck optical long-slit spectroscopy, and deep optical imaging (where available) for four galaxies. Our analysis reveals multiple possible reasons for the HI excess in these systems. One galaxy, AGC 10111, shows an HI disc that is counter-rotating with respect to the stellar bulge, a clear indication of external origin of the gas. Another galaxy appears to host a Malin 1-type disc, where a large specific angular momentum has to be invoked to explain the extremeMHI/* ratio of 166 per cent. The other two galaxies have early-type morphology with very high gas fractions. The lack of merger signatures (unsettled gas, stellar shells, and streams) in these systems suggests that these gas-rich discs have been built several Gyr ago, but it remains unclear how the gas reservoirs were assembled. Numerical simulations of large cosmological volumes are needed to gain insight into the formation of these rare and interesting systems.