How Many Pokémon Can You Get with Only Pokémon Yellow?
Assuming you do not use any glitches, the answer is 129. The Professor Oak Challenge Guide for Yellow can be found here, although I've made a shorter list of encounters below in case anything happens to the external guide.
The list of Pokémon you cannot get without trading, which equals 22 species, is as follows:
- Raichu (1) - 1 total
- You cannot evolve your starter Pikachu, and cannot get any other Pikachu, meaning Raichu is unobtainable without trading.
- Trade Evolutions (3) - 4 total
- Golem, Alakazam, and Gengar evolve by trade. Without the ability to trade, they are unobtainable. You can get Machamp via in-game trade.
- Two Eeveelutions (2) - 6 total
- As you only receive one Eevee, you can only receive one of its evolutions.
- One Fossil (2) - 8 total
- As you can only obtain one Fossil in Mt. Moon, you miss out on the other Fossil Pokémon and its evolution.
- One Hitmon (1) - 9 total
- As you can only obtain one Hitmon, you miss out on the other.
- Version Exclusives (12) - 21 total
- The following Pokémon, and their evolutions, cannot be obtained without the use of glitches or trading: Weedle, Ekans, Meowth, Koffing, Jynx, Electabuzz, Magmar.
- Mythical/Event (1) - 22 total
- Outside of glitches, Mew was only obtainable via events that are no longer active. However, Mew was never needed to get the certificate from the Game Freak employee.
Professor Oak Challenge Notes
Pre-Badge 1 (Brock)
- Pallet Town: Pikachu (1 total)
- Route 1: Pidgey, Pidgeotto, Pidgeot (4 total)
- Route 1: Rattata, Raticate (6 total)
- Route 22: Nidoran♂, Nidorino (8 total)
- Route 22: Nidoran♀, Nidorina (10 total)
- Route 22: Spearow, Fearow (12 total)
- Route 22: Mankey, Primeape (14 total)
- Viridian Forest: Caterpie, Metapod, Butterfree (17 total)
Badge 1 through Badge 2 (Misty)
- Route 3: Sandshrew, Sandslash (19 total)
- Route 4: Magikarp, Gyarados (21 total)
- Mt. Moon: Zubat, Golbat (23 total)
- Mt. Moon: Geodude, Graveler (25 total)
- Mt. Moon: Paras, Parasect (27 total)
- Mt. Moon: Clefairy (28 total)
- Cerulean City: Bulbasaur, Ivysaur, Venusaur (31 total)
- Route 24: Venonat, Venomoth (33 total)
- Route 24: Oddish, Gloom (35 total)
- Route 24: Bellsprout, Weepinbell (37 total)
- Route 24: Charmander, Charmeleon, Charizard (40 total)
- Route 5: Abra, Kadabra (42 total)
- Route 5: Jigglypuff (43 total)
- Route 11: Drowzee, Hypno (45 total)
- Diglett's Cave: Diglett, Dugtrio (47 total)
- Route 2: Mr. Mime: Trade Your Clefairy (48 total)
- Evolve Two Moon Stone Pokémon with Moon Stones from Mt. Moon (50 total)
Badge 2 through Badge 3 (Koga)
- Route 10: Machop (51 total)
- Route 10: Magnemite, Magneton (53 total)
- Rock Tunnel: Onix (54 total)
- Celadon City: Eevee & Evolution (56 total)
- Celadon Game Corner: Porygon (57 total)
- Celadon Game Corner: Vulpix (58 total)
- Buy a Fire Stone: Evolve Vulpix (59 total)
- Buy a Leaf Stone: Evolve Gloom (60 total)
- Buy a Leaf Stone: Evolve Weepinbell (61 total)
- Evolve Two Moon Stone Pokémon with Moon Stones from Rocket Hideout and Route 2 (63 total) (alternatively, purchase a Wigglytuff from Celadon Game Corner)
- Route 16: Doduo, Dodrio (65 total)
- Pokémon Tower: Gastly, Haunter (67 total)
- Pokémon Tower: Cubone, Marowak (69 total)
- Route 12: Machoke (& Machamp): Trade Your Cubone (71 total)
- Route 12/16: Snorlax (72 total)
- Route 17: Ponyta, Rapidash (74 total)
- Super Rod on Route 17: Shellder, Cloyster (76 total)
- Super Rod on Route 22: Poliwag, Poliwhirl, Poliwrath (79 total)
- Super Rod on Route 24: Goldeen, Seaking (81 total)
- Super Rod on Route 10: Krabby, Kingler (83 total)
- Super Rod on Route 12: Horsea, Seadra (85 total)
- Super Rod on Route 19: Staryu, Starmie (87 total)
- Super Rod on Route 19: Tentacool, Tentacruel (89 total)
- Route 13: Farfetch'd (90 total)
- Super Rod in Safari Zone: Dratini, Dragonair, Dragonite (93 total)
- Safari Zone: Exeggcute, Exeggutor (95 total)
- Safari Zone: Rhyhorn, Rhydon (97 total)
- Safari Zone: Tangela (98 total)
- Safari Zone: Chansey (99 total)
- Safari Zone: Scyther (100 total) (alternatively, purchase a Scyther from Celadon Game Corner)
- Safari Zone: Pinsir (101 total) (alternatively, purchase a Pinsir from Celadon Game Corner)
- Safari Zone: Tauros (102 total)
- Safari Zone: Kangaskhan (103 total)
- Saffron City: One Hitmon (104 total)
- Silph Co.: Lapras (105 total)
Badge 3 through Badge 4 (Lt. Surge)
- Surfing on Route 6: Psyduck, Golduck (107 total)
- Surfing on Route 12: Slowpoke, Slowbro (109 total)
- Power Plant: Voltorb, Electrode (111 total)
- Power Plant: Grimer, Muk (113 total)
- Power Plant: Zapdos (114 total)
- Cinnabar Lab: Aerodactyl (115 total)
- Cinnabar Lab: Fossil & Evolution (117 total)
- Pokémon Mansion: Growlithe, Arcanine (119 total)
- Pokémon Mansion: Ditto (120 total)
- Seafoam Islands: Seel, Dewgong (122 total)
Badge 4 through Badge 5 (Erika)
- Vermilion City: Squirtle, Wartortle, Blastoise (125 total)
Badges 5 through 8
- Seafoam Islands: Articuno (126 total)
Badge 8 through Elite Four
- Victory Road: Moltres (127 total)
Post-Elite Four
- Cerulean Cave: Lickitung (128 total)
- Cerulean Cave: Mewtwo (129 total)