There are many cards that trigger upon a successful run on HQ: Emergency Shutdown, Apocalypse et al, Desperado and Gabriel Santiago, Datasucker and others. All of these cards can benefit from a "double Inside Job" even without getting a random HQ access or ability to trash an upgrade.
However, one of the most infamous counters to the above and all of our other favorite HQ-accessing cards (cough Account Siphon cough) is the upgrade powerhouse Crisium Grid. Being five to trash is a nasty weight for most runners (outside of Drug Dealer / Faust, of course), but remember this little trick:
Our friend Feint's "if successful" trigger is a downside! Which means that when Crisium Grid prevents that trigger, you regain the ability to access, and therefore trash the Grid. Combine with an HQ interface that you probably should be considering if you're playing a heavy blue deck and you've got a nice little combo that compliments, rather than defines, your deck.