If I think about it in a (nearly) literal sense, the one tool I can't live without is my Fluke multimeter. There is no other tool that can perform as many functions or to the same degree. For example, a test light may tell you if there is voltage present or not, but won't identify a voltage drop or show fluctuations well. And that's only one of the many functions of a multimeter.
Runner up would be my smartphone. Along the same vein as the internet, it makes information so much easier to access and I don't have to run back and forth from my work to a computer monitor. I'm not *that* old, but I do remember doing everything on my first car out of a Haynes manual. Also helps to be able to call/text/email a friend or coworker if I get stuck with a real headscratcher, and to be able to send pics/audio of whatever is going on. Many other uses too, camera flash works as a flashlight, I occasionally reach it in tight places to take a pic of something that isn't visible, numerous apps that I use for work too (thread gauge, electrical reference with calculators, ruler, level, etc.) Not always the best tool for the job, but when quick-and-dirty will suffice it's there.
Excluding specialty tools that I may use once a week or less, I can't think of many tools whose function can't be done by another tool. Many make my job faster or easier, but can be (and in most cases have been) done with lesser tools. The current crop of 12v brushless tools has been a game changer, they've reached a point where their power, run time, compactness, and weight makes them suitable for many jobs and can be used all day comfortably, but I got by without them in the past. Same goes for ratcheting wrenches, a great innovation that saves time and effort, but I lived without them for years.