The vast majority of games are playable offline, there are some new ones that require you to be connected to the PSN though in order to combat the custom firmware 'threat'.
AFAIK all disc-based titles that have a single-player mode must be playable offline in order to be licenced for manufacture. If it tells you that you need to download anything to continue, try detatching the network cable and restarting the device so that it's purely offline.
This is correct, as I understand it. You can disable networking in the XMB for the same effect.