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How Many Pokémon Can You Get with Only Pokémon Silver?

Assuming you do not use any glitches, the answer is 206, or 207 if you have a Virtual Console release of Crystal, which allows you to get Celebi. The Professor Oak challenge guide for Crystal 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 52 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 (10) - 16 total
Alakazam, Machamp, Golem, Gengar, Politoed, Slowking, Steelix, Scizor, Kingdra, and Porygon2 evolve by trade. Without the ability to trade, they are unobtainable.
Version Exclusives (10) - 26 total
The following Pokémon cannot be obtained without the use of glitches or trading: Mareep, Flaaffy, Ampharos, Vulpix, Ninetales, Mankey, Primeape, Girafarig, Remoraid, and Octillery.
Red, Green, Blue, or Yellow Pokémon (17) - 43 total
Bulbasaur, Ivysaur, Venusaur, Charmander, Charmeleon, Charizard, Squirtle, Wartortle, Blastoise, Omanyte, Omastar, Kabuto, Kabutops, Articuno, Zapdos, Moltres, and Mewtwo can only be obtained from Red, Green, Blue, or Yellow.
Mythical/Event (2) - 44/45 total
Outside of glitches, Mew was only obtainable via events that are no longer active. Celebi can be obtained in the Virtual Console release of Crystal, but those playing the original cartridge release cannot obtain it.

Professor Oak Challenge Notes

Pre-Badge 1 (Falkner) - 46 Total Pokémon

Badge 1 through Badge 2 (Bugsy) - 78 Total Pokémon

To use Headbutt here before being able to use Cut, you will need to use Slowpoke, who learns Headbutt at level 34.

If your character has a Trainer ID ending in 7, you will not be able to get Heracross until Route 42.

Badge 2 through Badge 3 (Whitney) - 111 Total Pokémon

All of the Pokémon that can be hatched from the Odd Egg are available by breeding. Choosing to hatch Tyrogue simply means you have access to more Pokémon before Badge 3 than you otherwise would, since you can get four Pokédex entries.

Badge 3 through Badge 4 (Morty) - 143 Total Pokémon

Badge 4 through Badge 5 (Pryce) - 150 Total Pokémon

Badges 5 through Badge 7 (Chuck) - 154 Total Pokémon

Badge 7 through Badge 8 (Clair) - 170 Total Pokémon

Badge 8 through Elite Four - 174 Total Pokémon

Elite Four through Badge 16 - 203 Total Pokémon

Celebi can be obtained here if you have access to the 3DS Virtual Console release of Crystal. If you are able to obtain Celebi, the total for this section is 204.

Badge 16 through defeating Red - 206 Total Pokémon

If you are able to obtain Celebi, the total for this section is 207.