I try to finish every series that I pick up even if it takes me a year to watch it. However, one series that I never finished was Maria Sama ga Miteru. I liked the first season, but it was a bit too slow in the second season to where I haven't watched the rest of it. I didn't even know about the third season/ OVA until it was slated for American release.
Also, as much as I love Sailor Moon, I have yet to complete any arc beside Sailor Stars. Not because I find it boring, but because I took for granted that I could watch the other 4 arcs anytime or get the discs instead. But that didn't work out- even online for a while. I only put emphasis on watching Sailor Stars immediately because of the controversy around it in the States (Which it wasn't as shocking as the writeups as nudity, homosexuality, and gender changes aren't exactly new things to me and weren't when I watched it in '05, but one article I saw made it sound as if it reached ecchi to H levels or something).
Don't worry, I never finished Sailor Moon, either.
On the other hand, I heard that some Video on Demand sites are putting the original back on their lists and a little of it has to do with the remake coming up in January or February, so it may offer a second chance. That actually makes me happy, though I'm kinda mixed about whether to watch the old one in subs or dubbed. Dubs were what I started with, but when I tried to watch a fandub of Sailor Stars, I cringed. Not because the voice acting was that bad (albeit I wish that they kept the original Sailor Stars song or dubbed that instead of going for the song created by the English company AND there was no excuse to change the Starlights' names or even their band to the Twilights instead stare ), but the sub was what I saw for that one first and I liked the Japanese voices more.
I left many series, most because I found them boring, annoying, stupid, overrated and/or too long.
There are also some anime I dropped, but finished later, like for example Cowboy Bebop, because I needed a while to get into the story. It took me like 3 or 4 tries to watch all episodes, but in the end I was very touched and loved the show.
I dropped Naruto like the hot mess it is. I used to buy Shonen Jump just to read it, and I watched it every day it was on, but as the series continued on I just couldn't stand it. Bleach is the same way; it started out nice enough, but after so long a long series is just tiring.
Otherwise I try to finish all the anime I watch; most of the time they're short enough. Manga, however, is a different story.
I'm more of a manga person because I have spouts of ADD and need to jump from thing to thing, and I can't do that with anime because sooner or later I won't want to watch it when it's halfway through and I just can't bookmark it. Then i'd have to watch the whole beginning again.
I tend to drop a lot of anime because of this, and or i've gotten bored of it. But sooner or later I might find the spark to watch it again and then get hooked later.
It's been like that since reading One Piece.