like for example after you watch it it still haunts you you cant shake it off, but at the same time you are afraid of losing that feeling. You love it so much to the extend you download its ost and search where are the locations they shoot that drama/movie, you may fall in love with its characters or its plot or you still watch that movie/drama from time to time.

The first that made me feel this is taiyou no uta movie, our science teacher made us watch it at school...Thanks to her I became a fan of japanese cinema, became fan of yui and became fan of japanese countryside. But the drama the stuck in me the longest time is sekai no chusin(crying out loud in the center of the world). That thing made me depress for a month, the nostalgia factor of that drama gets me until now and I became fan of yamada cause of that. Movies like Departures, nobody knows, fish story, tegami, ozosaki no hiwari and just recently taisetsu no koto made me feel this too.

bambino - coz i watched it when i was having problems with work and it helped me a lot to feel good
like father, like son - it made me think on what i'll do and until now it's still making me think
oh, it's a hard question, because there are some dramas that affect me in different ways, until now...

Oshin; the very first drama I watched, it was like more than 20 years ago, affects me until now because I started to interested in Japan since then, but unfortunately, I don't really remember the story so I can't say I love the drama itself it that much

Nodame cantabile; is where I got idea for my online name:  soracantabile

Good Luck!. Maybe if I have to pick one, I will say this one.  The first time I watched this was about 10 years ago, and it changed my point of view about fairness until now. 

So if it's about the longest, maybe I'll say Oshin, but the deeper impact given by Good Luck!
For me, it has to be Koukou kyoushi 2003, it just got under my skin! Iki mo dekinai natsu was also one that I kept thinking about for a long time. 

And for me as well, Bambino! was one of those dramas that was hard to recover from. :) But it wasn't as "poerful" for me as the titles I just mentioned.
For me it was also the jdrama "Mother" because of its deep meaning. I think a lot about the woman's role as a mother and this drama meant a lot for me. Many people see being a mother as something so natural while I have trouble understanding it and even accepting the fact that someday I'll be a mother too.
For me it has a lot to do with music...sometimes a show OST comes to my mind when I'm not thinking about anything else, I start humming one of the songs inside my head and all the feelings I associate with the drama come back!
It happens most often with Rookies and Buzzer Beat,  I watched them at a time when I was wondering what to do with my life and if I should pursue my foolish dreams or not, and that's probably why their message stayed with me so much.
I thought I was the only one who had this kind of feeling. I have that mostly with japanese movies. I had that with BECK,  I Wish, Nobody Knows. These are I can come up with of the many I have seen that had an impact on me.
@soracantabile; i understand what you mean with that feeling. I have that with Hana-Kimi of 2006. My very first jdrama. I still get that feeling again when I hear the OST or see scenes on Youtube. Nostalgia feeling I think?

@ Toots and sakuraKhey; Mother was deeply moving. I could feel the connection between the young child and that woman, but also between that woman and her biological mother. By the way, I think you've seen the jdrama "Woman" as well?
Confessions (movie) and  Liar Game (dorama and movie), I love those movies and dramas wich are like puzzles they made me think about what is going to happen and how did it happen.

Departures (movie) is another one that made me think about what we believe is a decent job and made me see the beauty of funerals.

Byakuyako aka journey under a midnight sun- this is the darkest jdrama I have ever watched. I cannot watch it again. I couldn't stop watching it. I can still see it in my head and I haven't watched it in years. Everytime I hear the theme tune it's so beautiful but it makes me very sad. It made me think about how a bad decision you make as child can spread out like ripples in a pond and ruin the rest of your life. It made me hate all the adults in the drama. I never hated the main characters, I always saw them as victims of their circumstances and the adults around them.
Maou and Stand Up! are 2 dramas that I love and rewatch often. Maou gives me chills  and makes me fall in love with Ikuta Toma and Ohno Satoshi all over again. And the theme song by Arashi is just brilliant for this drama and it sets the tone for all that happens. Stand Up! is hilarious and brings me back to the days of my youth and innocence. Even though some the experience differ due to culture the feelings and memories are similar. Just reminds me of how at the time I thought life as a teenager was so complicated when really it wasn't. And it makes me wish I could go back to that time.
Midaregumo (1967)
3 - Iron (2004)
In The Mood For Love (2000)
Woman in the Dunes (1964)

If you're not a student of cinema or have an interest in the art of filmmaking, you won't get it.
It's amazing hoe some dramas and movies can stick with you for such a long time.

For me it is these three:

1 Litre of Tears - this is probably cliche, but I do think about this drama a lot and the actual girl who went through this back in the 70's and 80's. I first watched this drama when I was 14/15 and I am now turning 21. Because of this drama I have read Aya Kito's diary many times. 

Dear Friends - This is with one of my favorite actresses, Keiko Kitagawa. I actually watched this drama right after 1 Litre of Tears for the first time when I was 14. This affected me in a different way, it made me realize that you never know what another person is going through. 

And lastly....Hana Yori Dango - This was my very first Asian drama. I fell in love with the OST and pretty much everything. I also watched it when I was 14 and I still have all the Utada Hikaru and Arashi songs on my iPod. I have since watched it fully two more times. <3
N No Tame Ni - like seriously, it's over 2 months since i watched the series but up until now, remembering the scenes make me emotional. it affected me so much and i still have the urge to watch it again and again and again
1 Litre of Tears - cried tons and had deep discussions about the drama with my friends.

Stand Up! -  really moving drama about friendship and coming of age.

Mother - exceptional drama that made me go through a lot of emotions

:)