Is there any way, at all in any possible situation, to transfer Pokémon from Pokémon Red or Blue to Pokémon Ruby, Sapphire, or Emerald?

Imagine you have a Mew in your copy of Pokémon Red (through the immensely famous Mew Glitch on Route 25) and really want to get it to your copy of Pokémon Sapphire to complete your Pokédex. I have heard a lot about how you can't do this, and I am desperate to get it over. Is there any way possible, glitch, exploit, or anything to get a Pokemon over?

First and second generation games are completely incompatible with third generation and later games. There is no way to transfer a Pokémon from the original releases of Red and Blue to Sapphire and Emerald.

Without using external hardware and software, the other answers by Wrigglenite and Nolonar are correct: the Generation 1 and 2 game cartridges cannot trade with or transfer to the Generation 3 games by any normal means. I would normally leave it at that; however, you asked if it is possible through any means. To that, I say yes, though whether or not it is legitimate is down to the user's preference.

Tools exist that can extract a Game Boy (Color) cartridge's save file, such as the Submodule. Using this in combination with a save editor such as PKHeX, it is possible to transfer Pokemon from the Game Boy (Color) titles to the Game Boy Advance titles.