What's the problem with creating really large variables?
The problem that it takes space. You need to put this "large large" variable somewhere in your memory. Say there are 81M videos stored on YouTube, multiply this by the amount of bytes you spend on this large-large variable, and you get a huge waste. Almost all the videos (i.e., all but one) can work fine with 4-8 bytes for their watch counter, so why keep 1KB (say) for every watch counter? That's throwing away 81GB of space for no good reason!
Creating a variable-size variable (as suggested in Tushar's answer) is also an option, but then the overhead becomes time. Instead of having a simple register that holds this data, whose handling takes $\ll 10$ clock cycles per operation, a variable-size one may take $>10$ and even $>100$ clocks to handle per operation. Why waste this time, when only a small fraction of the vidoes hit the 1B watches?