Totally rely on it. I'm on a computer all day at work, cell phone, what have you. And same at home.
However, having said that, when Hurricane Sandy hit last year and we had no power for a week, I managed. All I wanted at that point was the heat to come on - I wasn't really caring much about the interwebz
I like it but when comes down I am fine with not net/cable/cell. With Sandy we did propane tank heater and managed fine. I did go to mcds a few times just to get on web and change of pace.
I don't have a computer, but I use my smartphone for so many things to stay connected to the internet for information & family. It seems like my 3rd hand!
One morning (and I think I posted about this that day), I came into work and....
plugged my Nook in to charge
plugged my phone in to charge
plugged my iPod into the computer to charge
The third thing I do when I get up in the morning is get online (first is the elliptical/treadmill, second is make my coffee)
I usually have my phone plugged in the car when I'm driving to keep it charged. And my phone has a really good battery (it's a Droid Razr Maxx HD, known for its longer battery life), but I always keep it plugged in anyway. Sometimes when Rick is working OT on Saturdays, he has to get out of bed around 3:30, so he'll use my phone as an alarm, just in case Bootser decides to sleep in that morning.
When we put the baking island in the kitchen, we're going to add a "charging nook" at the end of the island. It's just a small shelf with an electrical socket to plug all our stuff in to charge. The way it is now is that we have phones and Nook scattered about on the kitchen counters.