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Decided to crunch some numbers for Gen I (point out where I went wrong):

Each Pokemon can have a name of up to 12 characters long. Can be uppercase or lowercase letters (so, 26*2 = 52 characters), or 17 special characters, 52+17 = 69, can be represented in 7 bits. Means that a Pokemon's name equals 12*7 bits = 84 bits, or 10.5 bytes.

A Pokemon can be one of one hundred levels, can be represented as 7 bits.
Also saved is the amount of experience, highest level of experience is 1640000, can be represented as 21 bits, or 2.625 bytes.

A Pokemon may possess up to 4 moves. The best way to do this would be to store an indexed move list on the rom, and just save the 4 indices. Total number of moves in gen 1: 165 (give or take my miscount). Can be represented in 7 bits. 7 * 4 = 28 bits, or 3.5 bytes.

There are 151 kinds of Pokemon, an index representing which Pokemon it is can be stored in 8 bits, or 1 byte.

Means each Pokemon will take up 10.5 + 2.625 + 3.5 bytes + 1 byte = 17.625 bytes.

The player can hold 6 Pokemon in their party, as well as store up to 20 Pokemon in one of 12 boxes. Meaning that the player can store up to 6 + (20 * 12) = 246 Pokemon. 246 Pokemon * 17.625 bytes = 3105.75 bytes, or 3.03 kilobytes.

... and now I'm bored, I might come back to this later. Need to look at: player name, enemy name, play time, badges, position in world, position in story, money, backpack, seen Pokemon in Pokedex. Others? (did the game have optional side quests? Honestly cannot remember).