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 Eevee evolutions (2) - 6 total
- As you only receive one Eevee, you can only receive one of its evolutions.
- One Fossil Pokémon (2) - 8 total
- As you can only obtain one Fossil in Mt. Moon, you miss out on the other Fossil Pokémon (Omanyte or Kabuto) and its evolution.
- One of Hitmonlee/Hitmonchan (1) - 9 total
- As you can only obtain one of Hitmonlee or Hitmonchan, you miss out on the other.
- Version Exclusives (12) - 21 total
- The following Pokémon cannot be obtained without the use of glitches or trading: Weedle, Kakuna, Beedrill, Ekans, Arbok, Meowth, Persian, Koffing, Weezing, Jynx, Electabuzz, and Magmar.
- Mythical/Event (1) - 22 total
- Outside of glitches, Mew was only obtainable via events that are no longer active.
Professor Oak Challenge Notes
Pre-Badge 1 (Brock) - 17 Total Pokémon
- Gift in Pallet Town: Pikachu (1 total)
- Walking on Route 1: Pidgey, Pidgeotto, Pidgeot (4 total)
- Walking on Route 1: Rattata, Raticate (6 total)
- Walking on Route 22: Nidoran♂, Nidorino (8 total)
- Walking on Route 22: Nidoran♀, Nidorina (10 total)
- Walking on Route 22: Spearow, Fearow (12 total)
- Walking on Route 22: Mankey, Primeape (14 total)
- Walking in Viridian Forest: Caterpie, Metapod, Butterfree (17 total)
Badge 1 through Badge 2 (Misty) - 50 Total Pokémon
- Walking on Route 3: Sandshrew, Sandslash (19 total)
- Purchase on Route 4: Magikarp, Gyarados (21 total)
- Walking in Mt. Moon (all floors): Zubat, Golbat (23 total)
- Walking in Mt. Moon (all floors): Geodude, Graveler (25 total)
- Walking in Mt. Moon (B1F and B2F): Paras, Parasect (27 total)
- Walking in Mt. Moon (all floors): Clefairy (28 total)
- Gift in Cerulean City: Bulbasaur, Ivysaur, Venusaur (31 total)
- Walking on Route 24: Venonat, Venomoth (33 total)
- Walking on Route 24: Oddish, Gloom (35 total)
- Walking on Route 24: Bellsprout, Weepinbell (37 total)
- Gift on Route 24: Charmander, Charmeleon, Charizard (40 total)
- Walking on Route 5: Abra, Kadabra (42 total)
- Walking on Route 5: Jigglypuff (43 total)
- Walking on Route 11: Drowzee, Hypno (45 total)
- Walking in Diglett's Cave: Diglett, Dugtrio (47 total)
- In-game Trade on 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) - 105 Total Pokémon
- Walking on Route 10: Machop (51 total)
- Walking on Route 10: Magnemite, Magneton (53 total)
- Walking in Rock Tunnel (B1F): Onix (54 total)
- Gift in Celadon City: Eevee (55 total)
- Purchase in Celadon Game Corner: Porygon (56 total)
- Purchase in Celadon Game Corner: Vulpix (57 total)
- Buy a Fire, Water, or Thunderstone in Celadon Department Store: Evolve Eevee (58 total)
- Buy a Fire Stone in Celadon Department Store: Evolve Vulpix (59 total)
- Buy two Leaf Stones in Celadon Department Store: Evolve Gloom and Weepinbell (61 total)
- Evolve two Moon Stone Pokémon with Moon Stones from Rocket Hideout and Route 2 (63 total) (alternatively, purchase a Wigglytuff in Celadon Game Corner)
- Walking on Route 16: Doduo, Dodrio (65 total)
- Walking in Pokémon Tower (3F and above): Gastly, Haunter (67 total)
- Walking in Pokémon Tower (5F, 6F, and 7F): Cubone, Marowak (69 total)
- In-game Trade on Route 12: Machoke (& Machamp): Trade your Cubone (71 total)
- Static encounter on Route 12 or 16: Snorlax (72 total)
- Walking on Route 17: Ponyta, Rapidash (74 total)
- Fishing (Super Rod) on Route 17: Shellder (75 total)
- Buy a Water Stone in Celadon Department Store: Evolve Shellder (76 total)
- Fishing (Super Rod) on Route 22: Poliwag, Poliwhirl (78 total)
- Buy a Water Stone in Celadon Department Store: Evolve Poliwhirl (79 total)
- Fishing (Super Rod) on Route 24: Goldeen, Seaking (81 total)
- Fishing (Super Rod) on Route 10: Krabby, Kingler (83 total)
- Fishing (Super Rod) on Route 12: Horsea, Seadra (85 total)
- Fishing (Super Rod) on Route 19: Staryu (86 total)
- Buy a Water Stone in Celadon Department Store: Evolve Staryu (87 total)
- Fishing (Super Rod) on Route 19: Tentacool, Tentacruel (89 total)
- Walking on Route 13: Farfetch'd (90 total)
- Fishing (Super Rod) in the Safari Zone (all areas): Dratini, Dragonair, Dragonite (93 total)
- Walking in the Safari Zone (all areas): Exeggcute (94 total)
- Buy a Leaf Stone in Celadon Department Store: Evolve Exeggcute (95 total)
- Walking in the Safari Zone (Center and North): Rhyhorn, Rhydon (97 total)
- Walking in the Safari Zone (Center and West): Tangela (98 total)
- Walking in the Safari Zone (Center and East): Chansey (99 total)
- Walking in the Safari Zone (East and North): Scyther (100 total) (alternatively, purchase a Scyther in Celadon Game Corner)
- Walking in the Safari Zone (North and West): Pinsir (101 total) (alternatively, purchase a Pinsir in Celadon Game Corner)
- Walking in the Safari Zone (East and West): Tauros (102 total)
- Walking in the Safari Zone (North): Kangaskhan (103 total)
- Gift in the Fighting Dojo: One of Hitmonlee or Hitmonchan (104 total)
- Gift in Silph Co. (7F): Lapras (105 total)
Badge 3 through Badge 4 (Lt. Surge) - 122 Total Pokémon
Badge 4 through Badge 5 (Erika) - 125 Total Pokémon
Badges 5 through 8 - 126 Total Pokémon
Badge 8 through Elite Four - 127 Total Pokémon
Post-Elite Four - 129 Total Pokémon