I like to listen to podcasts while gaming quite a bit, so I deal with this issue a lot. I mostly buy cheap games I've played before on Steam (I already know what's going to happen, so I don't have to worry about story). Still, it's difficult to say what an acceptable amount of missing story is. The only thing I could really give ya is a list of everything I've played with podcasts and provide a few reasons why. Firing up Steam...
-Chime (music a lot of the experience, but after a while the music gets old.)
-Just Cause 2 (story in this was so stupid early on I decided I'd enjoy the game much more without sound anyway.)
-Need For Speed: Hot Pursuit (Story is really nonexistent, and podcasts are way better than EA trax).
BioShock 2 (second playthrough...)
Borderlands (again, playthrough 2, but I don't think you're missing much if you have subtitles on)
Burnout: Paradise (See NFS)
Jamestown (story's nice, but not necessary)
Resident Evil 5 (second playthrough, but you're not really missing much)
Super Meat Boy (again, not missing much)
Torchlight (See above)
Trine (See above)
Universe Sandbox (game literally has no sound)
It's a partial list, as I've omitted things that would probably fall under the BioShock 2/RE5 category, but that's what I've got. Hope you enjoy!