If you talk to the thirsty girl at athe Celadon Department store 100 times, she'll say "If you leave me alone, I'll give you a Flareth." And she gives you one. But what is a Flareth? I don't know...
If you collect 150 Pokémon and beat the game enough times, Prof. Oak will want to fight you. All of his Pokémon are at Level 80 or above. If you beat him, he will give you a Mew.
Mew is in the S.S. Anne.
Have nothing on your team but a Magikarp. At Lavender Town, rename it "Mew." It's now a Mew! (This does not work. I've tried it.)
Directly after beating the game, FLY to Celadon City and walk to Bill's house. If you talk to him ten times, he gives you a Pikablu.
Have 151 Pokémon. Put any four on your team, then add a Mewtwo and a Mew, both at level 100. Put them at the top of your team. FLY to Cinnabar Island. From there, SURF to Pallet Town. Go to Prof. Oak's Lab and talk to him. He should commend you for catching 151 Pokémon including a Mew and praise you for training them so good. Then he should tell you that the other day he caught a strange Pokémon and wants you to take care of it because you're such a good trainer. He gives you a Togepi at Level 5. It only knows two moves, Tackle and Withdraw.
If you beat the game 10 times without using a single item or ever healing, Prof. Oak will take you to a room where you fight to win the Pokégods.
If you use your Itemfinder near the Unknown Dungeon you will find a bottle with a message showing where to get Mew and Pikablu.
Have a Butterfree at Level 40. Then, talk to Mr. Psychic in Saffron to get the Psychic TM. Teach it immediately to your Butterfree. Leave the house and go to the Poké Center. Get three Geodude and three Moltres on your team. (Game Shark for the three Moltres, I guess.) Go back to Mr. Psychic and he will tell you how to get Mew.