How Many Pokémon Can You Get with Only Pokémon Let's Go Pikachu?
Assuming you do not use any glitches, the answer is 135. The Professor Oak challenge guide for Let's Go Pikachu can be found here, although I've made a Let's Go Pikachu-specific list of encounters below as well.
Let's Go Pikachu has a total of 153 available Pokémon: the standard Gen 1 151, as well as Meltan and Melmetal. The list of Pokémon you cannot get without trading, which equals 18 species, is as follows:
- Trade Evolutions (4) - 4 total
- Alakazam, Machamp, Golem, and Gengar evolve by trade. Without the ability to trade, they are unobtainable.
- Version Exclusives (11) - 15 total
- The following Pokémon cannot be obtained without the use of glitches or trading: Bellsprout, Weepinbell, Victreebel, Ekans, Arbok, Meowth, Vulpix, Ninetales, Pinsir, Koffing, and Weezing. (Persian is available as a gift Pokémon.)
- Mythical/Event (3) - 18 total
- Mew is available only with the Poké Ball Plus accessory. Meltan and Melmetal are available only via transfer from Pokémon Go, or by an event in China only.
Professor Oak Challenge Notes
Pre-Badge 1 (Brock) - 23 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 1: Oddish, Gloom (8 total)
- Walking on Route 22: Nidoran♂, Nidorino (10 total)
- Walking on Route 22: Nidoran♀, Nidorina (12 total)
- Walking on Route 22: Spearow, Fearow (14 total)
- Walking in Viridian Forest: Caterpie, Metapod, Butterfree (17 total)
- Walking in Viridian Forest: Weedle, Kakuna, Beedrill (20 total)
- Walking in Viridian Forest (Rare Spawn): Bulbasaur, Ivysaur, Venusaur (23 total)
Badge 1 through Badge 2 (Misty) - 51 Total Pokémon
Although not required for Pokédex completion, Alolan Rattata and Raticate can be obtained via an in-game trade in Cerulean City.
- Walking on Route 3: Sandshrew, Sandslash (25 total)
- Walking on Route 3: Mankey, Primeape (27 total)
- Walking on Route 3 (Rare Spawn): Charmander, Charmeleon, Charizard (30 total)
- Purchase on Route 4: Magikarp, Gyarados (32 total)
- Walking in Mt. Moon (all floors): Zubat, Golbat (34 total)
- Walking in Mt. Moon (all floors): Paras, Parasect (36 total)
- Walking in Mt. Moon (all floors): Geodude, Graveler (38 total)
- Walking in Mt. Moon (all floors): Clefairy (39 total)
- Walking in Mt. Moon (B2F): Clefable (alternatively, evolve Clefairy with a Moon Stone from Mt. Moon) (40 total)
- Walking in Mt. Moon (all floors): Onix (41 total)
- Walking in Mt. Moon (all floors, Rare Spawn): Chansey (42 total)
- Walking on Route 4: Psyduck, Golduck (44 total)
- Walking on Route 24: Venonat, Venomoth (46 total)
- Walking on Route 24 (Rare Spawn): Squirtle, Wartortle, Blastoise (49 total)
- Evolve Nidorino and Nidorina with Moon Stones from Mt. Moon (51 total)
Badge 2 through Badge 3 (any except Blaine or Giovanni) - 127 Total Pokémon
Although not required for Pokédex completion, some Alolan forms can be obtained via in-game trade: Geodude and Graveler in Vermilion City, Diglett and Dugtrio in Lavender Town, Sandshrew and Sandslash in Celadon City, Marowak in Fuchsia City, Raichu in Saffron City, and Grimer and Muk in Cinnabar Island.
- Walking on Route 5: Growlithe (52 total)
- Walking on Route 5: Abra, Kadabra (54 total)
- Walking on Route 5: Jigglypuff (55 total)
- Evolve Jigglypuff with a Moon Stone from Mt. Moon or Copycat's house: Wigglytuff (56 total)
- Gift in Vermilion City: Persian (57 total)
- Walking in Diglett's Cave: Diglett, Dugtrio (59 total)
- Walking on Route 11: Mr. Mime (60 total)
- Walking on Route 11: Drowzee, Hypno (62 total)
- Walking on Route 10: Krabby, Kingler (64 total)
- Walking in Rock Tunnel (all floors): Rhyhorn, Rhydon (66 total)
- Walking in Rock Tunnel (all floors): Cubone, Marowak (68 total)
- Walking in Rock Tunnel (all floors): Machop, Machoke (70 total)
- Walking in Rock Tunnel (all floors): Kangaskhan (71 total)
- Walking on Route 8: Arcanine (alternatively, purchase a Fire Stone in Celadon Department Store) (72 total)
- Walking on Route 7 (Rare Spawn): Porygon (alternatively, get one as a gift for defeating Giovanni in Saffron City) (73 total)
- Buy a Leaf Stone in Celadon Department Store: Evolve Gloom (74 total)
- Buy a Thunder Stone in Celadon Department Store: Evolve Pikachu from Viridian Forest (75 total)
- Walking in Pokémon Tower (3F and above): Gastly, Haunter (77 total)
- Static encounter on Route 12 or 16: Snorlax (78 total)
- Walking on Route 12: Farfetch'd (79 total)
- Walking on Route 14: Scyther (80 total)
- Walking on Route 14: Tauros (81 total)
- Walking on Route 16: Doduo, Dodrio (83 total)
- Walking on Route 17: Ponyta, Rapidash (85 total)
- Walking on Route 17: Eevee (86 total)
- Buy a Fire, Water, and Thunder Stone in Celadon Department Store: Evolve Eevee (89 total)
- Gift in Silph Co. (7F): Lapras (90 total)
- Gift in the Fighting Dojo: One of Hitmonlee or Hitmonchan (91 total)
- Sea Skimming on Route 22: Poliwag, Poliwhirl (93 total)
- Buy a Water Stone in Celadon Department Store: Evolve Poliwhirl (94 total)
- Sea Skimming on Route 6: Goldeen, Seaking (96 total)
- Sea Skimming on Route 10: Tentacool, Tentacruel (98 total)
- Sea Skimming on Route 10: Dratini, Dragonair, Dragonite (101 total)
- Sea Skimming on Route 12: Horsea, Seadra (103 total)
- Walking in the Power Plant: Voltorb, Electrode (105 total)
- Walking in the Power Plant: Magnemite, Magneton (107 total)
- Walking in the Power Plant: Grimer, Muk (109 total)
- Walking in the Power Plant: Electabuzz (110 total)
- Static encounter in the Power Plant: Zapdos (111 total)
- Sea Skimming on Route 19: Staryu, Starmie (alternatively, purchase a Water Stone in Celadon Department Store) (113 total)
- Walking in the Seafoam Islands (all floors): Jynx (114 total)
- Walking in the Seafoam Islands (all floors): Seel, Dewgong (116 total)
- Walking in the Seafoam Islands (all floors): Slowpoke, Slowbro (118 total)
- Sea Skimming in the Seafoam Islands (B3F and B4F): Shellder, Cloyster (alternatively, purchase a Water Stone in Celadon Department Store) (120 total)
- Static encounter in the Seafoam Islands (B4F): Articuno (121 total)
- Revive the Old Amber in Cinnabar Lab: Aerodactyl (122 total)
- Revive the Helix or Dome Fossil in Cinnabar Lab: One of Omanyte or Kabuto, and its evolution (124 total)
- Walking in Pokémon Mansion (all floors): Ditto (125 total)
- Walking in Pokémon Mansion (all floors): Magmar (126 total)
- Walking on Route 21: Tangela (127 total)
Badge 3 through Badge 7 (any except Blaine or Giovanni) - 129 Total Pokémon
Badge 7 through Elite Four - 131 Total Pokémon
Although not required for Pokédex completion, Alolan Exeggutor can be obtained via an in-game trade in Indigo Plateau.
- Walking in Victory Road (2F or 3F, Rare Spawn): The other of Hitmonlee or Hitmonchan (130 total)
- Static encounter in Victory Road (2F): Moltres (131 total)
Post-Elite Four - 135 Total Pokémon
- Walking in Cerulean Cave (all floors): Lickitung (132 total)
- Revive a Helix or Dome Fossil found in Cerulean Cave: The other of Omanyte or Kabuto, and its evolution (134 total)
- Static encounter in Cerulean Cave (B1F): Mewtwo (135 total)