Probably a Swans game, as they've been a lifelong second team. There is the 1996 PF, that 1999 early round game against North Melbourne, all those heart-stopping thrillers in the early Channel Ten days, the 2005 SF, 2010 finals series, the 2012 GF, etc. In a few instances I've supported Fremantle, Melbourne and (recently) GWS to beat them, but generally only the Eagles stands between me and the Swans.
That said, 2001 GF was a case where I hated Essendon and loved the Lions. But after half-time I had to go to a sleepover birthday, and other than listening to some of the fourth quarter on car radio, I missed the bulk of the second half and didn't know the result until the next morning. 2013 GF would have been similar, but I would have felt sad for Hawthorn if they came away from 2011-13 without a flag to show for it.
2002 Qualifying Final win by Collingwood, 2000 Qualifying Final win by Melbourne, Port's 2013 finals run, Adelaide's 2009 finals run, Freo in Sirengate (I was outraged!), etc. are other cases which stand out.