There are numbers out there that are so enormously, impossibly vast that to even write them down would require the entire universe. But here's the really crazy thing...some of these incomprehensibly huge numbers are crucial for understanding the world.
When I say "the biggest number in the universe", what I really mean is the biggest meaningful number, the largest possible number that is in some way useful. There are lots of contenders for this title, but I'll warn you now: there is a very real risk that trying to understand all this will blow your mind. But then, with extreme math, that's half the fun.
Googol and Googolplex
We might as well begin with what are quite probably the two largest numbers you've ever heard of, and are in fact the two largest numbers with commonly accepted definitions in the English language. (There's a fairly robust nomenclature available for naming numbers as high as you want to go, but you won't find these in dictionaries at the present time.) The googol, which has since become world famous (albeit misspelled) in the form of Google, began life in 1920 as a way to get children interested in large numbers.
To that end, mathematician Edward Kasner (pictured) took his two nephews, Milton and Edwin Sirotta, on a walk through the New Jersey Palisades. He asked them for any ideas they might have, and the then nine-year-old Milton proposed "googol." Where he got this particular word is unknown, but Kasner decided that 10^100 - or, the number one followed by a hundred zeroes - would henceforth be known as a googol.
But young Milton wasn't finished - he also proposed an even larger number, the googolplex. This number, according to Milton, was 1 followed by as many zeroes as you could write before you got tired. Though a charming idea, Kasner decided a more technical definition was needed. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition left open the dicey possibility that a random buffoon could become a greater mathematician than Albert Einstein simply by possessing greater endurance.
So, Kasner decided a googolplex would be 10^googol, or 1 followed by a googol of zeroes. To put that another way - and in similar notation to how we'll be dealing with various other numbers we'll be talking about - a googolplex is 10^10^100. To put that in some mindbending perspective, Carl Sagan once pointed out that it would physically impossible to write down all the zeroes in a googolplex, because there simply isn't enough room in the universe. If you filled the entire volume of the observable universe with fine dust particles roughly 1.5 micrometers in size, then the number of different combinations in which you could arrange and number these particles would be about one googolplex.
Linguistically speaking, googol and googolplex are probably the two biggest meaningful numbers (at least in English), but as we're about to find out, there's no end of ways to define "meaningful."
The Real World
If we're going to talk about the largest meaningful number, there's a not terrible argument that that really means we need to find the largest number with any real world significance. We can start the bidding with the current human population, which is currently about 6.92 billion. The global economy in 2010 is estimated to have been about $61.96 trillion, but both of those are dwarfed by the roughly 100 quadrillion cells that make up the human body. Of course, none of these can compare to the total number of particles in the universe, which is generally thought to be around 10^80 - a number so large that our language doesn't have an agreed upon word for it.
We can play around a bit with measurements as we get larger and larger - for instance, the weight of the Sun in tons will produce a smaller value than if you measure it in pounds. The fairest way to do this is to use the Planck units, which are the smallest possible measurements for which the laws of physics still hold. For instance, the age of the universe in Planck time is about 8 * 10^60. If we go right back to the first unit of Planck time after the Big Bang, we find the density of the universe was 5.1 * 10^96. We're getting bigger, but we still haven't even reached a googol.
The largest number with any real world application - or, in this case, real worlds application - is probably 10^10^10^7, which is one recent estimate of the number of universes in the multiverse. That number is so huge that the human brain would be literally unable to perceive all those different universes, as the mind is only capable of roughly 10^10^16 configurations. Actually, that number is probably the biggest with any practical application, assuming you don't buy into the whole multiverse idea. But there are still far larger numbers lurking out there. But in order to find them, we're going to need to venture in the realm of pure mathematics, and there's no better place to start than with the prime numbers.