I'm using a GNU/Linux phone, which feels like it gives me the best of both worlds. I have a full blown capable PC in my pocket should I need it, but it doesn't do anything that I don't want it to. I use it mostly in a pull-only manner - the only notifications I get are phone calls, SMS and IM messages from closest contacts. I can also receive notifications on work e-mail and work IM, but only when I explicitly leave the clients open, which I rarely do since I'm usually in front of a PC when I work anyway.

I think the only app that ever bothered me with unwanted notification was a backup app reminding about backups, with a big fat "don't remind me anymore" button attached to it ;)