I think the most valuable type of application to run in the background for me is a streaming radio app like Pandora or imeem Mobile. I hate that I cannot enjoy streaming content while surfing in Safari, writing emails, or checking out Twitter or Facebook. Looking through the screens of apps I have loaded on my iPhone, if I had a second choice (assuming my IM client was working with the upcoming Apple server technology) I think it would be a Twitter or Facebook app to allow me to stay current with status and updates.
If you could have one or two apps running in the background on your iPhone, what would you choose?