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How Many Pokémon Can You Get Without Evolving or Breeding in Pokémon HeartGold?

The player can get up to 373 species total (394 with the Pokéwalker) in HeartGold without trading, as described here. Without evolving any Pokémon, and only collecting evolutionary stages that the player can get in the wild, as a gift or purchase, or from an in-game trade, the player can get up to 268 species (287 with the Pokéwalker). Of the 373 possible species in HeartGold, the Pokémon unavailable without evolving are as follows:

Johto Starter Evolutions (2) - 2 total
The Starters are only available in their base forms, leaving their evolutions inaccessible.
Kanto Starter Evolutions (2) - 4 total
The Starters are only available in their base forms, leaving their evolutions inaccessible.
Hoenn Starter Evolutions (2) - 6 total
The Starters are only available in their base forms, leaving their evolutions inaccessible.
Stone Evolutions (24) - 30 total
Raichu, Nidoqueen, Nidoking, Clefable, Wigglytuff, Vileplume, Arcanine, Poliwrath, Victreebel, Cloyster, Exeggutor, Starmie, Vaporeon, Jolteon, Flareon, Bellossom, Sunflora, Ludicolo, Shiftry, Roserade, Mismagius, Honchkrow, Togekiss, and Gallade are inaccessible.
Baby Pokémon (6) - 36 total
Without the ability to breed, Pichu, Cleffa, Igglybuff, Wynaut, Happiny, and Mantyke are inaccessible.
Pokéwalker Baby Pokémon (6) - 42 total
Magby, Elekid, Smoochum, Mime Jr., Bonsly, and Munchlax can be obtained with the Pokéwalker, but are otherwise unavailable.
Other (63) - 105 total
Pidgeot, Hitmonlee, Hitmonchan, Espeon, Umbreon, Omastar, Dragonite, Crobat, Togetic, Ampharos, Azumarill, Forretress, Granbull, Magcargo, Piloswine, Octillery, Hitmontop, Blissey, Tyranitar, Mightyena, Swellow, Pelipper, Kirlia, Gardevoir, Slaking, Ninjask, Shedinja, Loudred, Exploud, Hariyama, Aggron, Camerupt, Grumpig, Flygon, Altaria, Crawdaunt, Claydol, Armaldo, Walrein, Salamence, Metagross, Staravia, Staraptor, Bibarel, Luxray, Wormadam, Mothim, Vespiquen, Cherrim, Ambipom, Lopunny, Gabite, Garchomp, Lucario, Hippowdon, Drapion, Toxicroak, Weavile, Lickilicky, Tangrowth, Yanmega, Gliscor and Mamoswine are inaccessible.
Pokéwalker Evolutions (8)
Even if the Pokéwalker is used, Delcatty, Sharpedo, Wailord, Milotic, Glalie, Froslass, Gastrodon, Lumineon, and Abomasnow still cannot be obtained. However, Bibarel can be obtained, making this a net loss of 8 for Pokéwalker players.

Professor Oak(-ish) Challenge Notes

This just contains a list of areas you can find each Pokémon, ordered as though you are playing a Professor Oak challenge and therefore need to obtain every Pokémon as early as possible.

Pre-Badge 1 (Falkner) - 19 Total Pokémon (55 with Pokéwalker)

By speaking to Primo in Violet City, the player can obtain a Slugma egg in exchange for a password. If you choose to get this egg, you can get Slugma for a total of 20 Pokémon.

With access to the Pokéwalker, the player can obtain the following Pokémon. Since Slowpoke can be obtained here, so can Headbutt encounters; you will have to train Slowpoke to level 25. The total with Pokéwalker access is 55.

Badge 1 through Badge 2 (Bugsy) - 35 Total Pokémon (58 with Pokéwalker)

To use Headbutt here before being able to use Cut, you will need to use Slowpoke, who learns Headbutt at level 25.

Badge 2 through Badge 3 (Whitney) - 53 Total Pokémon (69 with Pokéwalker)

Badge 3 through Badge 4 (Morty) - 77 Total Pokémon (88 with Pokéwalker)

Badge 4 through Badge 5 (Pryce) - 122 Total Pokémon (127 with Pokéwalker)

Badge 5 through Badge 7 (Chuck) - 124 Total Pokémon (129 with Pokéwalker)

Badge 7 through Badge 8 (Clair) - 131 Total Pokémon (136 with Pokéwalker)

Badge 8 through Elite Four - 136 Total Pokémon (140 with Pokéwalker)

Elite Four through Badge 9 (Misty) - 255 Total Pokémon (275 with Pokéwalker)

Several Pokémon need to be captured as both their base form and evolved form via the Safari Zone. I have attempted to create a method for getting through this as quickly as possible, but have not yet tested it. Only Pokémon for which the Safari Zone is required are listed.

First, set the six areas of the Safari Zone to Plains, Swamp, Rocky Beach, Mountain, Meadow, and Wetland.

In the Plains, you can get the following Pokémon immediately:

Swamp:

Rocky Beach:

Mountain:

Meadow:

Wetland:

After 70 total days:

Meadow:

Peak:

Wetland:

Wasteland:

After 80 total days:

Mountain:

Desert:

After 100 total days:

Wasteland:

Marshland:

Rocky Beach:

Savannah:

After 110 total days:

Forest:

Swamp:

Desert:

After 130 total days:

Savannah:

Peak:

Marshland:

The rest of the Pokémon in this section can be obtained normally in other locations.

With access to the Pokéwalker, the player can obtain the following Pokémon, bringing the total to 275. The three missing from the below list are Magby, Elekid, and Smoochum, which can be obtained before the very first Badge with the Pokéwalker but cannot be obtained otherwise.

Badge 9 through Badge 16 - 256 Total Pokémon (276 with Pokéwalker)

Badge 16 through defeating Red - 265 Total Pokémon (284 with Pokéwalker)

After defeating Red - 268 Total Pokémon (287 with Pokéwalker)