Wobbuffet's first appearance, under the ownership of Benny
Wobbuffet originally belonged to a Trainer named Benny. In Tricks of the Trade, Benny was searching for someone to trade Wobbuffet with for a stronger Pokémon at the Pokémon Swap Meet in Palmpona. Benny was not successful at first, though a Trainer with a Stantler seemed to like his offer, but then bailed for a Hoothoot at the last minute. Jessie's Poké Ball containing her Lickitung then accidentally fell into the machine instead when she crashed into it. Unbeknownst to her, Lickitung got swapped for Wobbuffet.
Jessie was unaware that her Lickitung had been traded to Benny for his Wobbuffet
Later, after the trade had commenced, Jessie wanted to send out her Lickitung to battle Ash and his friends, but to her surprise, Wobbuffet appeared instead. James then immediately looked up its moves for Jessie. Even though Wobbuffet had no offensive attacks, it managed to counter Chikorita's attacks back to her. Wobbuffet was blasted off along with the Team Rocket trio by Pikachu's Thunderbolt.
Since the episode Tunnel Vision, Wobbuffet had its place in the Team Rocket motto, first before Meowth (which usually left him very upset) and some episodes later, right after Meowth. It could often be heard crying out subsequent to the "Team Rocket is blasting off again!" line, even if it was not out of its Poké Ball when the team was sent blasting off. Wobbuffet was thus somewhat a fourth team member.
When Wobbuffet popped out of its Poké Ball in Power Play!, and redirected Meowth's attack back to himself, Jessie had the idea of building the Robo-Mecha Wobbuffet where it would counter any attack back to the user similar to Wobbuffet's Counter.
In Wobbu-Palooza!, Ash, his friends and Team Rocket came across a festival honoring Wobbuffet. Jessie's Wobbuffet popped out of its Poké Ball much to the villagers's excitement. The Wobbuffet Festival crashers then showed up to attack the Wobbuffet and Jessie sent out her Wobbuffet to battle them. Jessie's single-handedly took out the Wobbuffet Festival crashers's Pokémon with the use of Counter and Mirror Coat.
In Imitation Confrontation, Wobbuffet battled Duplica's Ditto, Mini-Dit. Mini-Dit transformed into Wobbuffet and used the same moveset to battle Wobbuffet and Arbok. Wobbuffet ended up losing to its miniature counterpart, however.
In The Wayward Wobbuffet, Wobbuffet was tasked with guarding the Poké Pod containing Ash's Pikachu. During the chase to get the Pod, Wobbuffet, along with the Pod and the key to it, was separated from Jessie and went on an action-packed adventure. It bumped into a thief named Goneff who was not happy with Wobbuffet bumping into his vehicle. However, Goneff took Wobbuffet hostage. Team Rocket then dug a hole and headed to the warehouse Goneff was hiding from underground to rescue Wobbuffet. Ash and his Totodile went with them only for Totodile to get badly hurt by Golem. When Golem was about to finish off Totodile, Wobbuffet jumped in the way and sent Golem flying to its Trainer. Ash was then able to get the Poké Pod back but Wobbuffet still had the key as it flew away along with the Team Rocket trio on the Wobbuffet-shaped balloon. Ash sent out Noctowl to get the key from Wobbuffet, however. Noctowl was successful and Pikachu was freed from the Poké Pod.
Advanced Generation series
In A Poached Ego!, Wobbuffet was sent out to battle Rico's Tyranitar. It was getting ready to counter Tyranitar's Hyper Beam but fainted as Hyper Beam was too strong.
In Claydol Big and Tall, Wobbuffet put on the disguise of a maid which made a humongous Claydol fall in love with it. This was part of a plan to get this dangerous Claydol back inside its stone Poké Ball. When Wobbuffet's disguise was revealed, Claydol was angry, but Claydol was returned to its stone Poké Ball before Wobbuffet was hurt.
Kanto Battle Frontier
Wobbuffet participated in its first Contest onscreen in A Hurdle for Squirtle. Jessie planned for it to use the countering technique against Harley's Ariados. However, this backfired, as Ariados's String Shot manipulated it into a puppet. Since then, it was used in other Contests, but would usually cause the elimination of Jessie for one reason or another.
Wobbuffet ran away from the group in Fear Factor Phony after a fake ghost made by some Ghost Pokémon scared it off. While it was running away it saw a Kirlia dancing and fell in love with it. When Team Rocket caught Pikachu and Kirlia, Wobbuffet jumped into the mecha after it and broke the Kirlia and its Psychic Pokémon friends free from Team Rocket's capture. Entirely thanks to Wobbuffet, Team Rocket's plan was completely ruined.
Diamond & Pearl series
Jessilina with her Wobbuffet
Wobbuffet went with Jessie, James, and Meowth to Sinnoh in Following A Maiden's Voyage! where it countered Meowth's Fury Swipes after an argument with Jessie. As a result, it sent Team Rocket blasting off and Pikachu falling in the forest. In One Big Happiny Family!, Wobbuffet participated in catching Brock's Happiny and accidentally tugged on Team Rocket's balloon after trying to pull her down. After the balloon fell to the ground, Wobbuffet, Meowth, and Mime Jr. made funny faces at Happiny but she was not happy about it. Later in the episode, Wobbuffet, Meowth, and Mime Jr. prevented Happiny from reuniting with Brock but she single-handedly defeated them by throwing them in the air.
In Staging a Heroes' Welcome! Jessie choose Wobbuffet for the appeals round in the Wallace Cup. After being sent out Jessie had Wobbuffet run straight towards her and use Counter, which sent her flying sky high. The judges were not impressed by this which caused Jessie to lose. In Double-Time Battle Training!, she sent out Wobbuffet and Yanmega to perform double moves by having Wobbuffet deflect Yanmega's Silver Wind with Mirror Coat. However, this failed as Team Rocket were sent blasting off.
Best Wishes series
Wobbuffet popping out behind, and reuniting with Jessie, James, and Meowth
Following the Team Rocket trio's journey in Sinnoh, Jessie was forced to leave Wobbuffet at Team Rocket's headquarters as it is not a Pokémon native in the Unova region and would attract attention.
Since then, Wobbuffet made appearances along with Meowth in the Pikachu shorts, PK23, Meloetta's Moonlight Senerade and Eevee & Friends. Strangely, the three of these shorts take place during the Best Wishes series, which Wobbuffet is normally absent from for the aforementioned reasons. It should be noted, however, that these shorts call into question their canonicity with the main anime series altogether.
Wobbuffet's only appearance in a Best Wishes series episode was in The Dream Continues!. After returning to Kanto and giving away their Unova Pokémon to Giovanni, the Team Rocket trio walked through a corridor only for Wobbuffet to pop up, surprising them. Wobbuffet was later seen with Team Rocket as they eavesdropped on Ash from the roof of his house in Pallet Town, marking its return to Jessie's team.
Wobbuffet reappeared as a member of Jessie's team in the XY series, where it was shown to be much more competent in battle. In Kalos, Where Dreams and Adventures Begin!, Wobbuffet was used in a battle against Pikachu, where it almost defeated Pikachu by returning his Thunderbolt right back at him with Mirror Coat. However, a wild Froakie came in and defended Pikachu from a reflected Electro Ball. Froakie helped defeat Wobbuffet by using his Frubbles to knock Wobbuffet on its back due to its inability to reflect moves that aren't attacks. Wobbuffet was soon blasted off by Pikachu, Froakie, and Clemont's Bunnelby. Even though Wobbuffet has been absent, it still been shown to be doing its comical pose every time Team Rocket do their motto.
Since its return, Wobbuffet has taken back its role as the "fourth member" of the team. It also seems to get more battling time now, and its capability of reflecting attacks has proven useful many times. Before Jessie caught her Pumpkaboo, Wobbuffet was her major battler, and did rather well, despite its incapability of using offensive moves.
Wobbuffet is used occasionally for battles. Wobbuffet's knowledge of moves like Counter and Mirror Coat comes in handy, since Team Rocket often finds itself on the receiving end of powerful attacks.
Wobbuffet usually serves as a comic relief character. It emerges from its Poké Ball
frequently as a running gag, and often at inappropriate times. Team Rocket has also had various vehicles and robots
based on it.
Wobbuffet popping out of its Poké Ball
Even though Jessie often seems annoyed with Wobbuffet, she was shown to truly care for it. This is showcased in a handful of episodes, including Who, What, When, Where, Wynaut?, in which Wobbuffet temporarily went missing. Jessie was very worried and desperately went searching for it on the island. Another instance was in Dealing With a Fierce Double Ditto Drama! where Jessie was upset at James's idea to send Wobbuffet to Giovanni since they couldn't tell it apart from Narissa's Ditto.
Wobbuffet also seems to care for Jessie deeply and see her as a motherly figure. This can be seen in The Wayward Wobbuffet, when it was lost and imagined a cloud in the shape of Jessie, childishly clapping with joy after noticing it. Even so, its loyalty has sometimes been in question, such as in Fear Factor Phony, where it temporarily left the group to hang out with some Psychic-type Pokémon and even single-handedly stopped Team Rocket's plan to steal Pokémon.