Honestly I think it was because during the first season of the Anime Ash was inexperienced as a Pokémon Trainer and at catching Pokémon.
Ash did have 6 Pokémon at one point in the first season of the Anime. But after Butterfree was released by Ash in EP021 he just didn't replace it with another Pokémon in his party.
The Haunter that he befriended in EP023 was not officially caught and was later left with Sabrina at the Saffron Gym, to keep her under control.
The Primeape Ash caught was left in the care of Anthony who offered to train Primeape for Ash, with the promise of training it to become a P1 champion.
Ash's Krabby that he caught in EP013 never made it into his actually party, may be because Ash already had a Water type in Squirtle, or may be because the Pokémon Creators just didn't want Ash to have Krabby in his side. So it would look epic when Krabby evolved into Kingler, in Ash's first battle in the Pokémon League and defeated all three of the opponents Pokémon.
None of the multiple Tauros that Ash and Brock caught in EP035 also made it into his party for unknown reasons, may be because he couldn't distinguish which Tauros to use or the Pokémon Creators just didn't want Ash to have Tauros in his party.
Ash's Muk that he caught in EP030 was sent to Professor Oak because Muk's stench was so rancid that it could seep through the Poké Ball.
You could say then why didn't he catch any other Pokémon?
Well as I mentioned above Ash was very inexperienced at the time, he was a step or seven behind other Pokémon trainers such as Gary. With Gary catching around forty-five Pokémon by the time Ash had only caught his seventh. To add to this Ash's Charizard would not listen to him or obey his commands in a Pokémon battle.
The first Pokémon Ash caught (Caterpie) was by luck as he didn't even battle it to capture it, he just threw a Poké Ball. When he tried the same technique with Pidgeotto it broke free, in fact if it wasn't for Misty telling him he had to battle it he probably would not of caught it. But of course what does Ash do when battling Pidgeotto, instead of sending out Pikachu to battle it, an Electric type he sends out his newly caught Caterpie a Bug type.
In addition to his lack of experience, many of Ash's first Gym Badges were given to him merely out of gratitude for his assistance rather than claiming an actual victory in battle over the Gym Leaders; of his first five Gym battles, only his victory over Lt. Surge was an actual victory.
Source: Knowledge and Bulbapedia
Hope I helped :)