After crossing over a certain point in Clannad: After Story I think everyone pretty much can't stop watching 'til the end, though that probably doesn't count since 3/4ths of Clannad/After Story are kind of boring in comparison.
I watched Cross Game and both seasons of Spice and Wolf weekly as they came out, which is very odd for me since I tend to wait for seasons to be over then grab the batches instead of watching them piecemeal.
I couldn't put Mawaru Penguindrum down because it kept taunting me, making me think that maybe, just maybe, in the end everything would come together and I would figure out what the heck I just watched. Surprise! It doesn't. I wasted 26 episodes of my life.
When I was in high school my anime club president handed me some DVDs with Narutaru AKA Shadow Star. I spent one sleepless night watching it from beginning to end and was deeply traumatized.
Both Sailor Moon and Detective Conan/Case Closed were addictive to me mostly because they are like the potato chip of anime. You can just keep popping in episodes with no real nutritional value and before you know it you've watched like 100 of them and can only vaguely remember the flavor.
I'm also addicted to showing people Higurashi. I don't tend to re-watch series but I've now seen this one about 5 times because every time I meet a new person who I think would like one of the best supernatural thriller stories of all time I sit them down and marathon it with them. Now I've grown a little more sane and just pester them to read the manga which has the advantage of filling in lots of the plot the anime glosses over and the fact that the art isn't absolute tripe like in the anime.
Oh and I watched a lot of Onegai My Melody (don't judge me!) because I couldn't get enough of this:
('Are you gonna read from the Kuromi note again?' 'Darn right I'm gonna!') The addiction dropped off when it got to Kuru Kuru Shuffle because the new villain had about 1/100th the charisma of the old one.