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It's not so much its Disney influences that make Ghibli popular so much as how its presented and advertised. Disney certainly did play a part, though, considering they're the guys that publish and dub the movies in the West.
But yeah, I've seen almost every Ghibli movie, even the non-Miyazaki ones. Love the everloving crap out of them. My favorite of them- Whisper of the Heart- was actually really set my eyes on pursuing gaming as a career. That movie means a lot to me.
But if I had to recommend ones that I think anyone in the thread would like, I'd say Spirited Away, Kiki's Delivery Service, and Howl's Moving Castle. Possibly Castle in the Sky and Ponyo, too. As for which ones I think are best, I'd say Spirited Away, Grave of the Fireflies, and Princess Mononoke. Save for a few (Arriety in particular comes to mind), most of them are great and I'd REALLY recommend you guys look into them more. Seriously, it's good stuff.