If Ash releases/trades/or gives away don’t count it
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Ash has obtained Pikachu, Caterpie, Metapod, Butterfree, Pidgeotto, Pidgeot, Charmander, Charmeleon, Charizard, Bulbasaur, Squirtle, Primeape, Haunter, Krabby, Kingler, Lapras, Muk, Tauros, Snorlax, Heracross, Totodile, Chikorita, Bayleef, Cyndaquil, Quilava, Noctowl, Torkoal, Treecko, Grovyle, Sceptile, Glalie, Tailow, Swellow, Phanpy, Donphan, Larvitar, Gible, Gligar, Gliscor, Buizel, Starly, Staravia, Staraptor, Chimchar, Monferno, Infernape, Turtwig, Grotle, Torterra, Sandile, Krokorok, Krookodile, Roggenrola, Boldore, Sewaddle, Swadloon, Leavanny, Tympole, Palpitoad, Scraggy, Snivy, Pidove, Tranquil, Unfezant, Tepig, Pignite, Oshawott, Froakie, Frogadier, Greninja, Fletchling, Fletchinder, Talonflame, Noibat, Noivern, Goomy, Sliggoo, Goodra, Aipom, Snorunt and Beedrill.
So, in total, if I counted correctly, Ash has obtained81 Pokemon.
Source
Ash has caught 80 species of Pokemon. (Including pre-evolved forms)
Source: Bulbapedia