That would be A LOT.
But I'll try.
There are 807 Pokemon
(Below lists how many alternate forms that a Pokemon has, not how many forms total that they have.)
Pikachu has 12, Pichu has 1, Unown has 27, Castform has 3, Deoxys has 3, Burmy has 2, Wormadam has 2, Cherrim has 1, Shellos has 1, Gastrodon has 1, Rotom has 5, Giratina has 1, Shaymin has 1, Arceus has 17, Basculin has 1, Darmanitan has 1, Deerling has 3, Sawsbuck has 3, Tornadus has 1, Thundrus has 1, Landorus has 1, Kyruem has 2, Keldeo has 1, Meloetta has 1, Genesect has 4, Vivillion has 19, Flabébé has 4, Floette has 5, Florges has 4, Furfrou has 9, Aeigslash has 1, Pumpkaboo has 3, Gourgeist has 3, Xerneas has 1, Hoopa has 1, Spinda has 4,294,967,294 (Thanks Astro!), Greninja has 1, Zygarde has 2, Silvally has 17, Oricorio has 3, Lycanroc has 2, Wishiwashi has 1, Minior has 7, Solgaleo has 1, Lunala has 1, Rattata has 1, Raticate has 1, Raichu has 1, Sandshrew has 1, Sandslash has 1, Vulpix has 1, Ninetales has 1, Diglett has 1, Dugtrio has 1, Meowth has 1, Persian has 1, Geodude has 1, Graveler has 1, Golem has 1, Grimer has 1, Muk has 1, Exeggutor has 1, Marowak has 1, Mimikyu has 1, Magearna has 1, and Necrozma has 3.
Total alternate forms: 4,294,967,498
(I am not counting the Nidorans because they have different Pokedex numbers.)
Venusaur, Butterfree, Rattata, Raticate, Pikachu, Raichu, Zubat, Golbat, Gloom, Vileplume, Kadabra, Alakazam, Doduo, Dodrio, Hypno, Rhyhorn, Rhydon, Rhyperior, Goldeen Seaking, Scyther, Scizor, Magikarp, Gyarados, Meganium, Ledyba, Ledian, Xatu, Sudowoodo, Politoed, Aipom, Ambipom, Wooper, Quagsire, Girafarig, Gligar, Steelix, Heracross, Sneasel, Weavile, Ursaring, Piloswine, Mamoswine, Octillery, Houndoom, Donphan, Torchic, Combusken, Blaziken, Beautifly, Dustox, Ludicolo, Nuzleaf, Shiftry, Meditite, Medicham, Roselia, Roserade, Gulpin, Swalot, Numel, Camerupt, Milotic, Cacturne, Relicanth, Starly, Staravia, Staraptor, Bidoof, Bibarel, Kricketot, Kricketune, Shinx, Luxio, Luxray, Combee, Pachirisu, Buizel, Floatzel, Gible, Gabite, Garchomp, Hippopotas, Hippowdon, Croagunk, Toxicroak, Finneon, Lumineon, Snover, Abomasnow, Tangrowth, Unfezant, Frillish, Jellicent, Pyroar, Meowstic, and Wobbuffet. (Thanks Seraphimon!)
Total Pokemon with gender differences: 98
Venusaur, Charizard Y, Charizard X, Blastoise, Alakazam, Gengar, Kangaskhan, Pinsir, Gyarados, Aerodactyl, Mewtwo Y, Mewtwo X, Ampharos, Scizor, Heracross, Houndoom, Tyranitar, Blaziken, Gardevoir, Mawile, Aggron, Medicham, Manectric, Banette, Absol, Garchomp, Lucario, Abomasnow, Beedrill, Pidgeot, Slowbro, Steelix, Sceptile, Swampert, Sableye, Sharpedo, Camerupt, Altaria, Glalie, Salamence, Metagross, Latios, Latias, Rayquaza, Lopunny, Gallade, Audino, and Diancie.
Total Pokemon with a Mega Evolution: 46
Groudon and Kyogre
Total number of Pokemon with a Primal Reversion: 2
The total for all of those categories combined together is 4,294,968,451.
Multiplied by 2 for shiny
**Minus three, as Castform's alternate forms cannot be shiny."
8,589,936,896 total currently.
Woah, that's a lot.
Notes: I did not count Rotom Dex, as you cannot catch a Rotom in Dex form. Also, Arbok's Gen 2 and R/S/E form was not counted. If you were to count both of these Pokemon, then the total would be 8,589,936,899. (Assuming that Rotom Dex cannot be shiny.)
Hope I helped!
Sources: This, this, this, this, this, and knowledge.