How Many Pokémon Can You Get in Legends: Arceus?
Legends: Arceus is unique among Pokémon games in that it is one of only a few games where all Pokémon can be obtained without trading with any other player. The game has a total of 242 Pokémon. Four of those - Phione, Manaphy, Darkrai and Shaymin - are not required in order to "complete the Pokédex" by the game's standards (catch Arceus or collect the Shiny Charm). While Phione and Manaphy can be easily obtained if the player knows how to unlock the Seaside Hollow, Shaymin and Darkrai both require the player to have save data of another Switch game on their system (Sword/Shield for Shaymin and Brilliant Diamond/Shining Pearl for Darkrai), meaning they may not be obtainable if the player does not have access to those other games.
The Professor Oak challenge guides (there are different 'difficulty levels', depending on whether the player wants to complete research tasks) for Legends: Arceus can be found here, although I've made a shorter list of encounters below in case anything happens to the external guide. This doesn't contain information about maximizing research tasks.
Note that all Pokémon can be obtained without the need for the free Daybreak update, apart from the aforementioned Shaymin/Darkrai. However, the addition of massive mass outbreaks in the Daybreak update makes it easier to get several species of Pokémon without needing to evolve them, though those aren't available until after completing Mission 18: The Counterpart.
Professor Oak Challenge Notes
Pre-Noble 1 (Kleavor) - 84 Total Pokémon
- Gift in Jubilife Village: One starter & its evolutions (3 total)
- Special encounter in Obsidian Fieldlands (Aspiration Hill): Bidoof (4 total)
- Special encounter in Obsidian Fieldlands (Aspiration Hill): Starly (5 total)
- Special encounter in Obsidian Fieldlands (Aspiration Hill): Shinx (6 total)
- Wild in Obsidian Fieldlands (Horseshoe Plains): Wurmple (7 total)
- Wild in Obsidian Fieldlands (Horseshoe Plains): Eevee, Espeon, Umbreon, Sylveon (11 total)
- Evolve Eevee with a Water Stone: Vaporeon (12 total)
- Evolve Eevee with a Thunder Stone: Jolteon (13 total)
- Evolve Eevee with a Fire Stone: Flareon (14 total)
- Evolve Eevee with a Leaf Stone or by going to the Mossy Rock in The Heartwood: Leafeon (15 total)
- Evolve Eevee with an Ice Stone: Glaceon (16 total)
- Wild in Obsidian Fieldlands (Horseshoe Plains): Ponyta, Rapidash (18 total)
- Wild in Obsidian Fieldlands (Horseshoe Plains): Drifloon (19 total)
- Wild in Obsidian Fieldlands (Horseshoe Plains): Mime Jr. (20 total)
- Wild in Obsidian Fieldlands (Horseshoe Plains): Buizel (21 total)
- Wild in Obsidian Fieldlands (Floaro Gardens): Pichu (22 total)
- Wild in Obsidian Fieldlands (Floaro Gardens): Silcoon, Beautifly (24 total)
- Wild in Obsidian Fieldlands (Floaro Gardens): Cascoon, Dustox (26 total)
- Wild in Obsidian Fieldlands (Deertrack Path): Kricketot (27 total)
- Wild in Obsidian Fieldlands (Deertrack Path): Geodude (28 total)
- Shaking trees in Obsidian Fieldlands (Deertrack Path): Burmy (29 total)
- Wild in Obsidian Fieldlands (Deertrack Path): Zubat (30 total)
- Wild in Obsidian Fieldlands (Deertrack Path): Floatzel (31 total)
- Wild in Obsidian Fieldlands (Deertrack Heights): Stantler, Wyrdeer (33 total)
- Wild in Obsidian Fieldlands (Deertrack Heights): Chimchar, Monferno, Infernape (36 total)
- Wild in Obsidian Fieldlands (Deertrack Heights): Staravia (37 total)
- Wild in Obsidian Fieldlands (Obsidian Falls): Pikachu (38 total)
- Evolve Pikachu with a Thunder Stone: Raichu (39 total)
- Air in Obsidian Fieldlands (Obsidian Falls): Gyarados (40 total)
- Air in Obsidian Fieldlands (Obsidian Falls): Staraptor (41 total)
- Wild in Obsidian Fieldlands (Windswept Run): Kricketune (42 total)
- Wild in Obsidian Fieldlands (Windswept Run): Abra (43 total)
- Wild in Obsidian Fieldlands (Windswept Run): Munchlax (44 total)
- Wild in Obsidian Fieldlands (Nature's Pantry): Paras, Parasect (46 total)
- Wild in Obsidian Fieldlands (Tidewater Dam): Bibarel (47 total)
- Wild in Obsidian Fieldlands (The Heartwood): Buneary, Lopunny (49 total)
- Wild in Obsidian Fieldlands (The Heartwood): Psyduck, Golduck (51 total)
- Wild in Obsidian Fieldlands (The Heartwood): Combee, Vespiquen (53 total)
- Shaking trees in Obsidian Fieldlands (The Heartwood): Cherubi, Cherrim (55 total)
- Wild in Obsidian Fieldlands (The Heartwood): Scyther (56 total)
- Evolve Scyther with a Black Augurite: Kleavor (57 total)
- Evolve Scyther with a Metal Coat: Scizor (58 total)
- Wild in Obsidian Fieldlands (The Heartwood): Golbat, Crobat (60 total)
- Wild in Obsidian Fieldlands (atop Oreburrow Tunnel): Machop, Machoke (62 total)
- Evolve Machoke with a Linking Cord: Machamp (63 total)
- Wild in Obsidian Fieldlands (atop Oreburrow Tunnel): Happiny, Chansey, Blissey (66 total)
- Wild in Obsidian Fieldlands (Grueling Grove): Graveler (67 total)
- Evolve Graveler with a Linking Cord: Golem (68 total)
- Water in Obsidian Fieldlands (Grueling Grove): Magikarp (69 total)
- Wild in Obsidian Fieldlands (Grueling Grove): Mothim (70 total)
- Wild in Obsidian Fieldlands (Grueling Grove): Heracross (71 total)
- Shaking trees in Obsidian Fieldlands (Sandgem Flats): Wormadam (72 total)
- Wild in Obsidian Fieldlands (Sandgem Flats): Snorlax (73 total)
- Wild in Obsidian Fieldlands (Sandgem Flats): Luxio, Luxray (75 total)
- Wild in Obsidian Fieldlands (Sandgem Flats): Drifblim (76 total)
- Wild in Obsidian Fieldlands (Sandgem Flats): Mr. Mime (77 total)
- Wild in Obsidian Fieldlands (Sandgem Flats): Kadabra, Alakazam (79 total)
- Wild in Obsidian Fieldlands (Sandgem Flats): Shellos, Gastrodon (81 total)
- Water in Obsidian Fieldlands (Sandgem Flats): Qwilfish, Overqwil (83 total) - you can lure Qwilfish with Oran Berries or other food
- Air in Obsidian Fieldlands (Sandgem Flats): Togekiss (84 total)
Pre-Noble 2 (Lilligant) - 138 Total Pokémon
- Space-time distortion in Obsidian Fieldlands: Sneasel (Johtonian), Weavile (86 total)
- Space-time distortion in Obsidian Fieldlands: Gengar (87 total)
- Space-time distortion in Obsidian Fieldlands: Steelix (88 total)
- Space-time distortion in Crimson Mirelands: Porygon, Porygon2, Porygon-Z (91 total)
- Wild in Crimson Mirelands (Golden Lowlands): Budew (92 total)
- Wild in Crimson Mirelands (Golden Lowlands): Carnivine (93 total)
- Wild in Crimson Mirelands (Golden Lowlands): Unown (94 total)
- Wild in Crimson Mirelands (Golden Lowlands): Gastly (95 total)
- Wild in Crimson Mirelands (Gapejaw Bog): Tangrowth (96 total)
- Wild in Crimson Mirelands (Gapejaw Bog): Croagunk (97 total)
- Water in Crimson Mirelands (Gapejaw Bog): Barboach, Whiscash (99 total)
- Wild in Crimson Mirelands (Gapejaw Bog): Teddiursa, Ursaring (101 total)
- Evolve Ursaring with a Peat Block: Ursaluna (102 total)
- Wild in Crimson Mirelands (Gapejaw Bog): Pachirisu (103 total)
- Wild in Crimson Mirelands (Gapejaw Bog): Roselia (104 total)
- Wild in Crimson Mirelands (Gapejaw Bog): Ralts (105 total)
- Wild in Crimson Mirelands (Gapejaw Bog): Haunter (106 total)
- Wild in Crimson Mirelands (Scarlet Bog): Onix (107 total)
- Wild in Crimson Mirelands (Scarlet Bog): Stunky, Skuntank (109 total)
- Wild in Crimson Mirelands (Scarlet Bog): Hippopotas, Hippowdon (111 total)
- Wild in Crimson Mirelands (Scarlet Bog): Rhyhorn, Rhydon (113 total)
- Evolve Rhydon with a Protector: Rhyperior (114 total)
- Wild in Crimson Mirelands (Cloudpool Ridge): Bonsly, Sudowoodo (116 total)
- Wild in Crimson Mirelands (Cloudpool Ridge): Roserade (117 total)
- Wild in Crimson Mirelands (Cloudpool Ridge): Murkrow, Honchkrow (119 total)
- Wild in Crimson Mirelands (Cottonsedge Prairie): Tangela (120 total)
- Wild in Crimson Mirelands (Cottonsedge Prairie): Petilil (121 total)
- Evolve Petilil with a Sun Stone: Lilligant (122 total)
- Wild in Crimson Mirelands (Cottonsedge Prairie): Togepi, Togetic (124 total)
- Wild in Crimson Mirelands (Droning Meadow): Yanma, Yanmega (126 total)
- Wild in Crimson Mirelands (Droning Meadow): Turtwig, Grotle (128 total)
- Wild in Crimson Mirelands (Shrouded Ruins): Lickitung, Lickilicky (130 total)
- Wild in Crimson Mirelands (Shrouded Ruins): Kirlia, Gardevoir (132 total)
- Evolve Kirlia with a Dawn Stone: Gallade (133 total)
- Wild in Crimson Mirelands (Holm of Trials): Goomy, Sliggoo, Goodra (136 total)
- Wild in Crimson Mirelands (Holm of Trials): Torterra (137 total)
- Wild in Crimson Mirelands (Holm of Trials): Toxicroak (138 total)
Pre-Noble 3 (Arcanine) - 174 Total Pokémon
- Space-time distortion in Cobalt Coastlands: Magnemite, Magneton, Magnezone (141 total)
- Wild in Cobalt Coastlands (Crossing Slope): Glameow, Purugly (143 total)
- Wild in Cobalt Coastlands (Aipom Hill): Aipom, Ambipom (145 total)
- Wild in Cobalt Coastlands (Deadwood Haunt): Chatot (146 total)
- Wild in Cobalt Coastlands (Deadwood Haunt): Duskull, Dusclops, Dusknoir (149 total)
- Water in Cobalt Coastlands (Sand's Reach): Remoraid (150 total)
- Wild in Cobalt Coastlands (Ginkgo Landing): Spheal, Walrein (152 total)
- Wild in Cobalt Coastlands (Ginkgo Landing): Skorupi, Drapion (154 total)
- Wild in Cobalt Coastlands (Castaway Shore): Octillery (155 total)
- Wild in Cobalt Coastlands (Veilstone Cape): Vulpix (Kantonian) (156 total)
- Evolve Vulpix (Kantonian) with a Fire Stone: Ninetales (Kantonian) (157 total)
- Wild in Cobalt Coastlands (Veilstone Cape): Growlithe (158 total)
- Evolve Growlithe with a Fire Stone: Arcanine (159 total)
- Wild in Cobalt Coastlands (Islespy Shore): Sealeo (160 total)
- Wild in Cobalt Coastlands (Islespy Shore): Piplup, Prinplup, Empoleon (163 total)
- Water in Cobalt Coastlands (Tranquility Cove): Mantyke, Mantine (165 total)
- Water in Cobalt Coastlands (Lunker's Lair): Tentacool, Tentacruel (167 total)
- Water in Cobalt Coastlands (Seagrass Haven): Finneon, Lumineon (169 total)
- Water in Cobalt Coastlands (Seagrass Haven): Basculin, Basculegion (171 total)
- Wild in Cobalt Coastlands (Firespit Island): Magby, Magmar (173 total)
- Evolve Magmar with a Magmarizer: Magmortar (174 total)
Pre-Noble 4 (Electrode) - 201 Total Pokémon
Pre-Noble 5 (Avalugg) - 218 Total Pokémon
Before Completing Mission 18: The Counterpart - 220 Total Pokémon
After Completing Mission 18: The Counterpart - 238 Total Pokémon