When Asriel and I were robbed last summer in Barcelona, there were a lot of documents and papers that were gone forever. Asriel, for instance, lost her credit card and driver's license, and we both lost our passports. Thankfully, the U.S. Consulate was on their A-game, and we were able to secure emergency passports in less than a day.
On my end, though, much of the process was sped along because I had memorized some key numbers. For emergencies, it will help tremendously if you have these numbers tucked away, either in the back of your mind or written down in a separate location:
Passport number: When filling out any emergency paperwork, being able to summon this number from memory makes the process go that much more smoothly—especially when your passport is missing!
Social security & driver's license: Like your passport number, these digits also make the process of verifying your identity during an emergency much, much easier. It can also mean the difference between friends and family being contacted in a timely fashion, if the need arises.
Airline confirmation numbers: In the event of a lost boarding pass, having a confirmation code will likely endear you to the counter agents, who are used to dealing with less-prepared passengers. Working with someone who has their crap together is a relief to them, and they'll be more enthusiastic about assisting you.
Any international calling codes you might be using: From contacting hosts/hotels ahead of time to calling home, knowing the calling code of your countries of stay could mean the difference in keeping a room reservation in the event of your running late.
Emergency phone numbers in visiting countries: Do you know what to dial if you have an emergency in Egypt? Sorry, it's not 911. Check WikiTravel or the U.S. Bureau of Consular Affairs for emergency numbers of countries on your itinerary. But while we're here, Egypt's emergency number is 122 (or 126 for the tourist police).
Can you think of anything I've left out?