The Master
Bronze Squire
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One of the primary reasons I like anime is that most of the shows are serialized. Some take it too far and stretch stupid shit over multiple episodes (DBZ being the most famous example), or have filler episode that totally break the arc you're in and frustrate you if you're watching week-to-week (or, worse, filler arcs that are multiple episodes long). But anime has a ton of excellent series that fit here.
Full Metal Alchemist: Brotherhood
Princess Tutu
Trigun
Cowboy Bebop+the movie to finish it off
Legend of Galactic Heroes
Those are all finished, with basically no filler, and are all really good and draw you in.
Also for shows that haven't been mentioned, Twin Peaks, though it is unfinished, is very much in this vein. Nothing makes sense if you don't watch each episode and it actually helps a lot to go back and rewatch it from the beginning when you get to the end.
Full Metal Alchemist: Brotherhood
Princess Tutu
Trigun
Cowboy Bebop+the movie to finish it off
Legend of Galactic Heroes
Those are all finished, with basically no filler, and are all really good and draw you in.
Also for shows that haven't been mentioned, Twin Peaks, though it is unfinished, is very much in this vein. Nothing makes sense if you don't watch each episode and it actually helps a lot to go back and rewatch it from the beginning when you get to the end.