Emergency Essentials
Trash Bags
Pardon the ick-factor here, but if the plumbing is down, you'll want a way to deal with waste in a sanitary way. Ready.gov suggests keeping trash bags with plastic ties for handling any human waste along with a stash of toilettes or hand sanitizer. Also, if you have any family members who aren't potty-trained, make sure you have a few days' worth of diapers handy as well.
Blankets
Heat can become a hot commodity when the power goes out. At home, keep a sleeping bag (maybe a couple, depending on the number of people in your family) with your emergency food stash. At the office or in the car, keep a warm blanket. Even if it's summer or you live in a warm climate, you'll want warmth once the sun goes down.
Cash
Guess what won't be at the top of your list in the event of a disaster? Finding an ATM (assuming you can find one that's working). And since credit card machines will likely be down, it won't hurt to have some spare cash on hand. Keep it in a waterproof/fireproof box with your important documents. For your car or office, store your cash in something like an iWallet, which requires your fingerprint to unlock. (Plus, it never hurts to have some cash in your car anyway, in case you are running on empty and forgot your wallet at home.)
Extra Gas
Let's say you need to get out of town as fast as possible. Unfortunately, you're low on gas and, judging from the gas station lines, it looks like everyone else in the city is too. Keep a container of gas in your trunk in case a moment like this were ever to occur (regardless, it's helpful if you were ever to run out of gas on the road). Or, an equally effective tip is to get in the habit of never allowing your gas tank to get more than halfway empty at any time.
Walking Shoes
Especially at work and in the car, keep a pair of walking shoes in your emergency kit or bag. In the event that you have to evacuate your office or that your car breaks down, wearing heels or going barefoot are two options you don't want to decide between.
Cell Phone Charger
As much as the cellphone has become a daily necessity, an emergency can turn it into a lifeline. Always keep an extra phone charger in your car. It will come in handy even when the emergency is that your cellphone died and you need to let daycare know you'll be late picking up the kids. For home or office, you may want to consider getting a solar-powered cell phone charger, since a regular one wouldn't be useful during a power outage.
