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The Pokemon, or mons, that have the lowest encounter rate overall in all gens that they are available. Sorry, this might take a while too.

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And, if there's any pokemon that you only are gifted(I think Porygon is one), don't count those either.
I can't access YouTube, or Bulbapiedia.
Can we ignore Pokemon with 0% encounter chance? If a Pokemon has multiple encounter chances in the same game, like 10% on one route and 15% on the next route, what do we do with those Pokemon?
Yes, you can, because you can't even get those, and do the average if they have more than one location. pecentage on location 1 plus percentage on location 2, etc, divided by number of locations

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Generation I

Since Red/Blue and Yellow have very different catch lists and percentages, I have divided this list between those two games. In this generation, none of the fishing rods have a 1% chance to get anything, so all of the following Pokemon are encountered either in Grass, or by Walking in caves or buildings.

Red/Blue

There were a lot of 1% encounter Pokemon in Red and Blue. Most of these are evolved Pokemon that are very rare in one version, but a lot more common in the other. Doing this did mean that these games had (theoretically) less version exclusives than later entries, but some of the Pokemon that were available were a lot harder to get in one version than the other. The main exceptions to this is, of course, the nightmarish task of catching certain Safari Zone encounters (especially Chansey!).

Caterpie: Route 25, Grass, Red Only
Chansey: Safari Zone (Central Area), Walking
Clefairy: Mountain Moon (1F), Walking
Dewgong: Seafoam Islands (B3F), Walking
Ditto: Cerulean Cave, Walking
Golbat: Seafoam Islands (B4F), Walking
Golduck: Seafoam Islands (1F and B2F), Walking, Red Only
Graveler: Victory Road (2F), Walking
Haunter: Pokemon Tower (3F only), Walking
Kangaskhan: Safari Zone (Western Area), Walking
Kingler: Seafoam Islands (B1F), Walking, Blue Only
Marowak: Victory Road (1F), Walking
Muk: Pokemon Mansion (all floors), Walking, Red Only
Pinsir: Safari Zone (Eastern Area), Walking, Blue Only
Scyther: Safari Zone (Eastern Area), Walking, Red Only
Seadra: Seafoam Islands (B1F), Walking, Red Only
Slowbro: Seafoam Islands (1F and B2F), Walking, Blue Only
Tauros: Safari Zone (Northern Area), Walking
Weedle: Route 25, Grass, Blue Only
Weezing: Pokemon Mansion (all floors), Walking, Blue Only
Yellow

Yellow had less 1% encounters than Red and Blue, but this is largely because it is only a single game, and so avoided the "version exclusive" encounters of the other Generation I games.

Chansey: Safari Zone (Central Area), Walking
Clefairy: Mountain Moon (1F), Walking
Dodrio: Route 17, Grass
Fearow: Route 9, Grass
Pidgeotto: Routes 24 and 25, and Viridian Forest, Grass
Pinsir: Safari Zone (Northern Area), Walking
Raticate: Route 11, Grass
Scyther: Safari Zone (Eastern Area), Walking
Slowbro: Seafoam Islands (B2F), Walking
Tangela: Safari Zone (Western Area), Walking
Venomoth: Routes 14 and 15, Grass
Generation II

Generation II did add several new methods to encounter Pokemon (Rock Smashing rocks and Headbutting trees), but none of these gave encounter rates of 1%. Several new mechanics that altered encounter rates were introduced, however - Time of Day and Swarms - and these did affect Pokemon with a 1% encounter rates. For Time of Day, the Pokemon in question were generally a lot common at another time of day (e.g. Quagsire and Zubat are both more common at Night). Swarms are more interesting, as that meant that at certain times Pokemon with very low encounter rates were a lot more common for a short period of time. Of the 1% Pokemon, this could affect Yanma, Dunsparce, Marill, and Snubbull, so those Pokemon's encounter rates assume that there is no swarm of them at the time.

Gold and Silver

Aerodactyl: Route 14, Trade
Technically, I am cheating by including Aerodactyl on this list, since it is a trade, and so a definite encounter. However, you need to trade a Chansey for it, and since Chansey is so rare, it may as well be a 1% encounter!
Chansey: Routes 13, 14, 15, Grass
Dunsparce: Dark Cave (Violet City side), Walking
Marill: Mt. Mortar (Entrance), Walking
Quagsire: Route 26, Grass, Morning Only
Snubbull: Route 38, Grass
Yanma: Route 35, Grass
Zubat: Route 32, Grass, Morning Only
Crystal

Aerodactyl: Route 14, Trade
Technically, I am cheating by including Aerodactyl on this list, since it is a trade, and so a definite encounter. However, you need to trade a Chansey for it, and since Chansey is so rare, it may as well be a 1% encounter!
Chansey: Routes 13, 14, 15, Grass
Dunsparce: Dark Cave (Violet City side), Walking
Jynx: Ice Path (1F and B1F), Walking, Morning and Day Only
Pupitar: Mt. Silver (Summit), Walking, Day Only
Sneasel: Ice Path (B1F), Walking, Night Only
Weezing: Burned Tower (B1F), Walking
Yanma: Route 35, Grass
Generation III

Ruby and Sapphire

Electrode: New Mauville (Basement), Walking
Illumise: Route 117, Grass, Sapphire Only
Kecleon: Routes 118, 119, 120, 121, 123, Grass
Magneton: New Mauville (Basement), Walking
Surskit: Routes 102, 111, 114, 117, 120, Grass/Surfing
Tentacruel: Abandoned Ship, Surfing
Volbeat: Route 117, Grass, Ruby Only
Wailord: Route 129, Surfing
Emerald

Note that Area 5 of the Safari Zone is the area to the far southeast, and is only accessible after entering the Hall of Fame.

Electrode: New Mauville (Basement), Walking
Goldeen: Routes 102, 111, 114, 117, 120, Surfing
Kecleon: Routes 118, 119, 120, 121, 123, Grass
Magneton: New Mauville (Basement), Walking
Nuzleaf: Route 114, Grass
Octillery: Safari Zone (Area 5), Super Rod
Quagsire: Safari Zone (Area 5), Surfing
Seedot: Routes 102, 117, 120, Grass
Tentacruel: Abandoned Ship, Surfing
Volbeat: Route 117, Grass
Wailord: Route 129, Surfing
FireRed and LeafGreen

Note that for Unown certain letters only have a 1% chance of spawning. Specifically, these are the '?' '!' and 'O' Unown symbols.

Chansey: Safari Zone (Entrance Area), Grass
Clefairy: Mt. Moon (1F), Walking
Dragonair: Safari Zone (All Areas), Super Rod
Kakuna: Routes 24 and 25, Grass, LG Only
Koffing: Celadon City, Surfing
Golduck: Seafoam Islands (B3F and B4F), Surfing, FR Only
Grimer: Celadon City, Super Rod
Haunter: Pokemon Tower (3F), Walking
Kangaskhan: Safari Zone (Area 3), Grass
Lapras: Icefall Cave (Back Cave), Surfing
Metapod: Routes 24 and 25, Grass, FR Only
Nidoran-female: Route 3, Grass, FR Only
Nidoran-male: Route 3, Grass, LG Only
Pinsir: Safari Zone (Area 1), Grass, LG Only
Psyduck: Routes 4, 10, 11, 12, 13, 19, 20, 21, 24, Pallet Town, Cerulean City, Vermilion City, Vermilion Harbor, Cinnabar Island, One Island, Treasure Beach, Kindle Road, Bond Bridge, Icefall Cave (Back Cave), Five Isle Meadow, Memorial Pillar, Resort Gorgeous, Five Island, Water Labyrinth, Water Path, Green Path, Outcast Island, and Tenoby Ruins, Super Rod, FR Only
Scyther: Safari Zone (Area 1), Grass, FR Only
Slowbro: Seafoam Islands (B3F and B4F), Surfing, LG Only
Slowpoke: Routes 4, 10, 11, 12, 13, 19, 20, 21, 24, Pallet Town, Cerulean City, Vermilion City, Vermilion Harbor, Cinnabar Island, One Island, Treasure Beach, Kindle Road, Bond Bridge, Icefall Cave (Back Cave), Five Isle Meadow, Memorial Pillar, Resort Gorgeous, Five Island, Water Labyrinth, Water Path, Green Path, Outcast Island, and Tenoby Ruins, Super Rod, LG Only
Tauros: Safari Zone (Area 2), Grass
Unown: Tanoby Chambers, Walking
Wobbuffet: Cerulean Cave (B1F), Walking
Generation IV

Several new methods of catching Pokemon were added in this generation, as well as the return of the previous capture methods. The ones that have 1% chance to encounter a Pokemon are Honey Trees and the Poke Radar. The Poke Radar is only given after getting the National Dex. Honey is a very annoying method. There are 21 Honey Trees in the game, but 4 will have a different encounter list than the other 17. To differentiate between these (when required) the 4 different Honey Trees will be called "Honey Tree (Special)". Which Honey Trees are special will be randomly chosen for every new game, however. The Honey Tree Pokemon with a 1% chance will have no locations given, since they can be encounter in any Honey Tree or Special Honey Tree.
Swarms again return, as well, which still increases the chance of encountering some of the listed Pokemon for a short period of time.

Diamond and Pearl

Azurill: Great Marsh (Areas 1, 2, 3, 4, and 6), Grass
Bidoof: Great Marsh (Areas 1, 2, 3, 4, and 6), Grass
Heracross: Honey Trees (Normal)
Kirlia: Routes 203 and 204, Poke Radar
Munchlax: Honey Trees (Special)
Shellos: Route 224, Grass
Platinum

Beautifly: Route 205 and Eterna Forest, Grass
Chimecho: Mt. Coronet (5F), Walking
Dustox: Route 205 and Eterna Forest, Grass
Gastrodon: Route 213, Pastoria City, and Canalave City, Surfing
Golbat: Oreburgh Gate (B1F), Walking
Heracross: Honey Trees (Normal)
Munchlax: Honey Trees (Special)
Pelipper: Routes 219, 220, 221, 222, and Sunnyshore City, Surfing
Tentacruel: Routes 205 and 212 (South), Surfing
HeartGold and SoulSilver

Chansey: Routes 13, 14, and 15, Grass
Dunsparce: Dark Cave (Violet City Entrance), Walking
Marill: Mt. Mortar (Entrance), Walking
Snubbull: Route 38, Grass
Yanma: Route 35, Grass
Generation V

There are very few 1% encounter Pokemon in either of the Unova games. Aside from those mentioned below, most encounters have a minimum of 5% chance to encounter (or 2.5% chance if you want to take into account the gender differences of Pokemon like Jellicent or Unfezant). In addition to the 1% encounters below, Tynamo is the only other one with a lower encounter chance (2% on several of the floors of Chargestone Cave).

I think the lack of rare encounters is due to the introduction of the "special" encounter methods. Whenever you are are near grass or water, or in a cave, there is a small chance of rustling grass, rippling water, or dust clouds, respectively, appearing. These "special" encounters contain different Pokemon than are normally found on the route. If a Pokemon in a "special" encounter has a 5 or 10% chance to appear, then you could still be searching for it for a very long time, so it really may as well be a 1% chance.

Seasons may also change the encounter rate for some Pokemon (usually just between Winter and the rest of the seasons). Every month is a new season (so the cycle doesn't match the actual seasons in the real world).

Black and White

Chinchou: Route 18, Fishing
Cryogonal: Twist Mountain, Walking, Spring, Summer, and Autumn Only
Dragonite: Dragonspiral Tower (Outside), Fishing in Rippling Water
Black 2 and White 2

Dragonite: Dragonspiral Tower (Outside), Fishing in Rippling Water
Mienfoo: Dragonspiral Tower (Entrance), Dark Grass, Winter Only

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Thanks G. Guy!
Thanks but sorry for not posting Gen six afterwards
It's okay, I realize how much it is, just as long as it eventually gets there, I'm fine with it.
What your source for this information?
Knowledge ( Bulbapedia)