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I think final showdown of Ash and Gary is one of the best for me.
For those who haven't watched these, possible spoilers ahead:
Spoiler:
I really enjoyed the part when Ash's Charizard is having a showdown against Gary's Blastoise, that was so intense, not to mention that Charizard turning the match into a fiery steamy battlefield and turned it to Ash's victory. I would still enjoy watching the match no matter if Ash was winning or losing that time because of how good the battle was.
My choice is a little different. I think that Ash's best battle was the one between him & Lt. Surge way back in Season 1, coz I think that's when Ash & Pikachu actually started working together, as a team :)
Another of my favorite is Ash's Gym Battle with Pryce in Season 5, especially the battle between Pikachu & Piloswine. I really liked the way Pikachu hung on, despite his type disadvantage :)
Ash vs. Paul, though I still have some issues with it, such as Infernape being way too overpowered to the point where it made Ash's other Pokemon look weak. Torterra should have been the one to take down Aggron for example; Ash would have still made his point if Infernape only KO'd Ninjask and Electivire.
Ash vs Blaine was one of my favourites, I used to watch that battle on repeat whilst studying in my uni's library. XD Ash's championship match in the Orange League was another favourite: "The challenger wins the match! The challenger wins the match!" *liquid pride every time* :P