One way to solve this problem is to give every computer in the world a unique identifier. In this hypothetical past, when there were only 10 computers, we could just label each computer with a number 1, 2, 3, etc. and then we could use that number to uniquely talk about each computer. Maybe my friend owns computer 3, he tells me his computer is 3, and on my computer which is 4, I say I want to connect to computer 3. So my computer looks at every computer connected to it and finds the one that reports itself as computer 3. This is pretty simple, and actually would work great if it weren't for a few fundamental problems:

1. There are more than 10 computers in a world. Way too many for a human brain to remember each unique number. Too many for my computer to ask every other connected computer what number it is as it tries to find a particular computer.
2. Humans are not as good at remembering "random" series of numbers as they are at remembering names and words in their own language. I can remember 10 random English words easier than I can 10 random numbers between 0-255, and that's not a fair comparison, because there are more English words than 255. Also some names are more valuable than others, because they are easier to remember, or represent more popular concepts. For example more people may want their computer to be named #1 rather than #2425, because #1 is easier to remember and people have a preference for being #1 in a competitive sense.
3. People all speak different languages, and it's easier to remember words and names in the language(s) you know. This is especially complex if my friend and I speak different languages. His computer might have a name in his language that is hard for me to remember. We need some system where we can use a translation of his name into a name that I can understand and remember.
4. Say my friend buys a new computer between telling me about the game and me trying to download it. This new computer might have a different number than his old one, but it's still **his** computer, and he likely means for me to get the game from his **new** computer rather than his old one. It would be nice if he could tell the world "I got a new computer. Please use computer 9 now, not computer 3" without having to communicate with me specifically.
5. Files and information transfer hold value, so someone might want to pretend to be my friend's computer. How can I be sure I got my friends file, not someone else who tried to pretend to be my friend? I somehow need some guarantees that the information my friend told me is enough to confidently route me to his computer, not someone else's.
6. The file on my friends computer might be in some folder. Once I'm connected to his computer, how do I find the file I am looking for? This is sort of a sub-problem, since my friend could include the information about which folder it's in as he's telling me the ID of his computer, but we'll likely want to treat this as a sub-problem, since it's likely easier to solve the problem of finding the right file **after** we solve the problem of connecting to the correct computer.
7. There might be a **lot** of people that want this file. Maybe my "friend" is actually Taylor Swift, and the "game" is actually a new album she just finished. Millions of people want to download this album but a single computer is not powerful enough to serve all of those people at the exact same time. For a variety of hardware reasons, we can't have a single physical computer that everyone is downloading from. We need to set up a system where the load of transferring the file is distributed amongst a collection of computers that all have the file. Even if everyone is finding the file using the same information that Taylor told them.
8. Related to #6 Each individual computer may break, or it may go offline for one reason or another. We can't have one computer failure cause the entire system to be inaccessible. We need to have a dynamic system where people get routed to a different computer when the one they were trying to connect to is unavailable
9. People live all over the world, and downloading a file from France while I live in Canada is not always that fast. It would be nice if my friend could manage a computer that is not physically located at his house, so that people that he can share files quickly with people who do not live close to him.