The dominant source class at very high energies (VHE; > 1 TeV) is well-known to be highly-aligned jetted AGN, or blazars. Nearly all of these are low-power high-synchrotron-peaking (HSP) BL Lac objects, thought to be the aligned counterparts of FR I radio galaxies. In contrast, the powerful quasar-type blazars and low-synchrotron peaking (LSP) BL Lacs are very rarely seen at VHE, presumably because their inverse Compton emission peaks in the MeV range. A notable exception is the source AP Librae, a moderately powerful LSP BL Lac, which has a remarkable ‘Compton’ spectrum ranging over 10 orders of magnitude, from soft X-rays to TeV energies. The TeV detection by H.E.S.S. in 2010 prompted the suggestion that the VHE emission in this source could be due to inverse-Compton upscattering of the CMB by the X-ray emitting kiloparsec-scale jet. We have recently obtained HST infrared imaging showing that TeV emission via IC/CMB can be strongly ruled out due to the lack of IR emission required from the kpc-scale jet. However, we show that the Fermi/LAT lightcurve of this source is unusual, being consistent with a steady emission which has only occasionally been punctuated by flaring events. We propose that the kpc-scale X-ray and steady Fermi/LAT emission is indeed due to IC/CMB, but the TeV emission remains unexplained.