Pokémon Red Version, Blue Version, and Yellow Version - Encountering Mew
The following method lets you encounter the Pokémon Mew at level 7, in In Pokémon Red Version, Pokémon Blue Version, and Pokémon Yellow Version. It was discovered sometime before July 2003 and came to my attention by then.
The Method
This method requires that you haven’t battled two trainers: the Youngster with a Level 17 Slowpoke, on the way to Bill’s house; and the Gambler with two Poliwags and a Poliwhirl, facing the Underground Path east of Saffron City.
You also need a Pokémon that knows Fly, as well as whatever you need to catch the Mew.
To begin with, stand at the door facing the Gambler and save the game. Then move one step closer, and as quickly as you can, press START just when you move one step closer and before he can challenge you to a battle. If you’re successful, the menu will appear. Without leaving the menu, fly to Cerulean City. Don’t save the game after this, until you’ve successfully captured Mew.
You will now be unable to use the Start, A, or B buttons.
In Cerulean City, walk to the Youngster, keeping as far as you can from him and making sure he walks to your character. (The buttons should function again.) Defeat him, and fly to Saffron City.
In Saffron City, go to the gate east of the city. The menu will appear. As soon as it appears, press B, and you will encounter a Mew.