Platinum Only

Assuming you do not use any glitches and do not have the ability to get the encounters that require a GBA Pokémon cartridge inserted, the answer is 374. The Professor Oak challenge guide for Platinum can be found here, although I've made a shorter list of encounters below in case anything happens to the external guide.

With the ability to get the GBA dual-slot encounters, you can get an additional 32 Pokémon, bringing the total to 406.

The list of Pokémon you cannot get without trading or using dual-slot encounters, which equals 374 species, is as follows:

Two Starters (6) - 6 total: As you can only select one starter Pokémon, the other two starters, and their evolutions, are unavailable; this totals 6 Pokémon.
Trade Evolutions (16) - 22 total: Alakazam, Golem, Machamp, Gengar, Rhyperior, Dusknoir, Porygon2, Porygon-Z, Scizor, Electivire, Magmortar, Politoed, Slowking, Kingdra, Huntail, and Gorebyss evolve by trade. Without the ability to trade, they are unobtainable.
One Fossil Pokémon (2) - 24 total: As you can only obtain one of the Skull or Armor Fossil in the Underground, you miss out on the other Fossil Pokémon (Cranidos or Shieldon) and its evolution.
Spiritomb (1) - 25 total: Spiritomb requires the player to be able to connect with other players in the Underground. Without multiplayer access, it is unobtainable.
Version Exclusives (12) - 37 total: The following Pokémon cannot be obtained without the use of glitches or trading: Misdreavus, Mismagius, Murkrow, Honchkrow, Glameow, Purugly, Stunky, Skuntank, Trapinch, Vibrava, Flygon, and Clamperl.
Other Region Starters (27) - 64 total: All Kanto, Johto, and Hoenn starters must be transferred from Gen 3 or traded from HeartGold or SoulSilver.
Other Region Legendaries (14) - 78 total: All Legendaries from Kanto, Johto, and Hoenn (except for Articuno, Zapdos, and Moltres) must be transferred from Gen 3 or traded from HeartGold or SoulSilver.
Regigigas (1) - 79 total: Regigigas is only obtainable if the player has Regirock, Regice, and Registeel in their party. As those are locked to transfer or trade, Regigigas is thus unobtainable.
Dual Slot (31) - 110 total: The following Pokémon cannot be obtained without the use of the GBA dual-slot encounters: Caterpie, Metapod, Butterfree, Weedle, Kakuna, Beedrill, Ekans, Arbok, Sandshrew, Sandslash, Vulpix, Ninetales, Growlithe, Arcanine, Pineco, Forretress, Shuckle, Teddiursa, Ursaring, Lotad, Lombre, Ludicolo, Seedot, Nuzleaf, Shiftry, Sableye, Mawile, Zangoose, Seviper, Lunatone, and Solrock. (The 32nd Pokémon is Gengar, which was mentioned separately.)
Mythical/Event (9) - 119 total: Outside of glitches, Mew, Celebi, Jirachi, Deoxys, Phione, Manaphy, Darkrai, Shaymin, and Arceus were only obtainable via events that are no longer active or are difficult to come by.

Professor Oak Challenge Notes

Pre-Badge 1 (Roark) - 36 Total Pokémon

Badge 1 through Badge 2 (Gardenia) - 76 Total Pokémon

Badge 2 through Badge 3 (Fantina) - 90 Total Pokémon

Badge 3 through 4 (Maylene) - 139 Total Pokémon

Badges 4 through 6 (Byron) - 157 Total Pokémon

Badges 6 through 7 (Candice) - 169 Total Pokémon

Badges 7 through Elite Four - 179 Total Pokémon

Post-Elite Four - National Pokédex - 374 Total Pokémon

All National Pokédex Pokémon are only available after defeating the Elite Four.

If the player has access to dual-slot capabilities, the following Pokémon can also be obtained after obtaining the National Pokédex:

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How Many Pokémon Can You Get with Only Pokémon Platinum?