  The cores of ultra-luminous infrared galaxies (ULIRGs) are very dense
environments, with a high rate of star formation and supernova explosions. They
are thought to be sites of cosmic-ray acceleration, and are predicted to emit
$\gamma$-rays in the GeV to TeV range. So far, no ULIRG has been detected in
$\gamma$-rays. Arp 220, the closest ULIRG to Earth, has been well studied, and
detailed models of $\gamma$-ray production in this galaxy are available. They
predict a rather hard $\gamma$-ray spectrum up to several TeV. Due to its large
rate of star formation, high gas density, and its close proximity to Earth, Arp
220 is thought to be a very good candidate for observations in very-high-energy
(VHE; 100 GeV - 100 TeV) $\gamma$-rays. Arp 220 was observed by the VERITAS
telescopes for more than 30 hours with no significant excess over the
cosmic-ray background. The upper limits on the VHE $\gamma$-ray flux of Arp 220
derived from these observations are the most sensitive limits presented so far
and are starting to constrain theoretical models. We also present upper limits
for the VHE flux from the ULIRG IRAS 17204-0014, the starburst galaxy IC 342,
and the active galaxy 3C321.
