Berserk: The Golden Age- I remember when I first picked Berserk up, and how hard it was to get into. The art style and the dark fantasy setting were interesting, but they really weren't enough to sell it for me. Then, we found out about Guts' past. His time with the Band of the Hawk was a fine mix of character bonding, some impressive fight scenes and drama.
Fairy Tail: Phantom Lord- I actually almost gave up on Fairy Tail halfway into Galuna Island. But then, once that arc had finished, I was surprised by how invested I'd become in the series once Natsu and Co. came back to find the guild under attack. It had a a lot of my favorite fights and character designs in the series, and I really like how it established Lucy as part of Fairy Tail.
If I talked about One Piece, you'd be here all day reading, so I'll just talk about one of my favorite arcs; Drum Island. For me, it had the most emotional depth to it, at least in the first half of the Grand Line. While it definitely still had its share of funny moments, I really felt the urgency of Luffy having to get Nami to a doctor.Then there's Chopper...His flashback was one of the most crushing for me. Dalton deserves a special mention, too. I feel people rarely talk about him, but he was a pretty strong character, whose past helped shape the kingdom's future.