Err medium...
At present I own a Xperia X10, an I know people don't really like Sony Ericsson for some odd reason, but I got accustom over the years I guess. It's my first smart phone and since I got it, I viewed the market. I won't say apps made me/influence me to get the phone but it has truely enhanced my experience with it. Few games here and there, not really for me but my smaller cousins and stuff; Adobe to read pdfs, app to kill processes...doesn't come with a file manager, app for that.
May seem a lot when I actually check but I won't really rate the influence as High as I'm not really addicted to the apps.
Also to add to my 'medium' rating, I would actually recommend a phone with decent apps, prefablely any Andriod based, since I don't really like BlackBerry's market much and not familiar with Apple's market.